6715:,. The underlying idea of separability is to make a countable set of points of the index set determine the properties of the stochastic process. Any stochastic process with a countable index set already meets the separability conditions, so discrete-time stochastic processes are always separable. A theorem by Doob, sometimes known as Doob's separability theorem, says that any real-valued continuous-time stochastic process has a separable modification. Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line.
1830:-dimensional Euclidean space or other mathematical spaces, where it is often interpreted as a random set or a random counting measure, instead of a stochastic process. In this setting, the Poisson process, also called the Poisson point process, is one of the most important objects in probability theory, both for applications and theoretical reasons. But it has been remarked that the Poisson process does not receive as much attention as it should, partly due to it often being considered just on the real line, and not on other mathematical spaces.
50:
8751:-dimensional Euclidean space or some manifold. In general, a random field can be considered an example of a stochastic or random process, where the index set is not necessarily a subset of the real line. But there is a convention that an indexed collection of random variables is called a random field when the index has two or more dimensions. If the specific definition of a stochastic process requires the index set to be a subset of the real line, then the random field can be considered as a generalization of stochastic process.
9804:, for a continuous-time stochastic process with any metric space as its state space. For the construction of such a stochastic process, it is assumed that the sample functions of the stochastic process belong to some suitable function space, which is usually the Skorokhod space consisting of all right-continuous functions with left limits. This approach is now more used than the separability assumption, but such a stochastic process based on this approach will be automatically separable.
9260:
1665:
9305:, and then become more popular after Bachelier's thesis was translated into English in 1964. But the work was never forgotten in the mathematical community, as Bachelier published a book in 1912 detailing his ideas, which was cited by mathematicians including Doob, Feller and Kolmogorov. The book continued to be cited, but then starting in the 1960s, the original thesis by Bachelier began to be cited more than his book when economists started citing Bachelier's work.
9793:, is to assume that the stochastic process is separable. Separability ensures that infinite-dimensional distributions determine the properties of sample functions by requiring that sample functions are essentially determined by their values on a dense countable set of points in the index set. Furthermore, if a stochastic process is separable, then functionals of an uncountable number of points of the index set are measurable and their probabilities can be studied.
741:
20231:
8797:. There are different interpretations of a point process, such a random counting measure or a random set. Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic process, though it has been remarked that the difference between point processes and stochastic processes is not clear.
19844:
9471:, in a less mathematically rigorous way than Kolmogorov, while studying Brownian movement. The differential equations are now called the Kolmogorov equations or the KolmogorovâChapman equations. Other mathematicians who contributed significantly to the foundations of Markov processes include William Feller, starting in the 1930s, and then later Eugene Dynkin, starting in the 1950s.
8230:
the next value, then it holds for all future values. The exact mathematical definition of a martingale requires two other conditions coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes. Martingales are usually defined to be real-valued, but they can also be complex-valued or even more general.
1807:, the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randomly occurring in certain time windows.
355:
9116:, the latter adopting the term martingale for the stochastic process. Methods from the theory of martingales became popular for solving various probability problems. Techniques and theory were developed to study Markov processes and then applied to martingales. Conversely, methods from the theory of martingales were established to treat Markov processes.
8150:, that have the Markov property, which means the next value of the Markov process depends on the current value, but it is conditionally independent of the previous values of the stochastic process. In other words, the behavior of the process in the future is stochastically independent of its behavior in the past, given the current state of the process.
9526:
distributions. For example, both the left-continuous modification and the right-continuous modification of a
Poisson process have the same finite-dimensional distributions. This means that the distribution of the stochastic process does not, necessarily, specify uniquely the properties of the sample functions of the stochastic process.
1555:, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say,
1779:
zero to some given time is a
Poisson random variable that depends on that time and some parameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process.
7491:
791:. Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge, particularly due to the index set being uncountable. If the index set is the integers, or some subset of them, then the stochastic process can also be called a
7903:
4758:
8229:
A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value. In discrete time, if this property holds for
5029:
can be interpreted as time, a stochastic process is said to be stationary if its finite-dimensional distributions are invariant under translations of time. This type of stochastic process can be used to describe a physical system that is in steady state, but still experiences random fluctuations. The
1778:
The
Poisson process is a stochastic process that has different forms and definitions. It can be defined as a counting process, which is a stochastic process that represents the random number of points or events up to some time. The number of points of the process that are located in the interval from
1763:
The Wiener process is a member of some important families of stochastic processes, including Markov processes, LĂ©vy processes and
Gaussian processes. The process also has many applications and is the main stochastic process used in stochastic calculus. It plays a central role in quantitative finance,
9529:
Another problem is that functionals of continuous-time process that rely upon an uncountable number of points of the index set may not be measurable, so the probabilities of certain events may not be well-defined. For example, the supremum of a stochastic process or random field is not necessarily a
9504:
In mathematics, constructions of mathematical objects are needed, which is also the case for stochastic processes, to prove that they exist mathematically. There are two main approaches for constructing a stochastic process. One approach involves considering a measurable space of functions, defining
9374:
derived the
Poisson distribution when developing a mathematical model for the number of incoming phone calls in a finite time interval. Erlang was not at the time aware of Poisson's earlier work and assumed that the number phone calls arriving in each interval of time were independent to each other.
9159:
Although
Khinchin gave mathematical definitions of stochastic processes in the 1930s, specific stochastic processes had already been discovered in different settings, such as the Brownian motion process and the Poisson process. Some families of stochastic processes such as point processes or renewal
8172:
with a countable state space (thus regardless of the nature of time), but it has been also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space). It has been argued that the first definition of a Markov chain, where
1786:
The homogeneous
Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process. If the parameter constant of the Poisson process is replaced with some non-negative
9403:
who studied Markov chains in the early 20th century. Markov was interested in studying an extension of independent random sequences. In his first paper on Markov chains, published in 1906, Markov showed that under certain conditions the average outcomes of the Markov chain would converge to a fixed
8409:
Martingales mathematically formalize the idea of a 'fair game' where it is possible form reasonable expectations for payoffs, and they were originally developed to show that it is not possible to gain an 'unfair' advantage in such a game. But now they are used in many areas of probability, which is
8126:
is used when discussing and assuming certain conditions for a stochastic process to resolve possible construction issues. For example, to study stochastic processes with uncountable index sets, it is assumed that the stochastic process adheres to some type of regularity condition such as the sample
5945:
is also used, however some authors use the term version when two stochastic processes have the same finite-dimensional distributions, but they may be defined on different probability spaces, so two processes that are modifications of each other, are also versions of each other, in the latter sense,
9234:
published, in 1919 and 1921, work where he studied the probability of a symmetric random walk returning to a previous position in the lattice. PĂłlya showed that a symmetric random walk, which has an equal probability to advance in any direction in the lattice, will return to a previous position in
8905:
contributed significantly to the field, more specifically, to the kinetic theory of gases, by presenting work where he modelled the gas particles as moving in random directions at random velocities. The kinetic theory of gases and statistical physics continued to be developed in the second half of
9390:
studied the counting problem and derived
Poisson probabilities as a solution to a family of differential equations, resulting in the independent discovery of the Poisson process. After this time there were many studies and applications of the Poisson process, but its early history is complicated,
8417:
Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference. They have found applications in areas in probability theory such as queueing theory and Palm
6283:
is a property of a stochastic process based on its index set in relation to the probability measure. The property is assumed so that functionals of stochastic processes or random fields with uncountable index sets can form random variables. For a stochastic process to be separable, in addition to
945:
in
English with its current meaning, which relates to chance or luck, date back to the 16th century, while earlier recorded usages started in the 14th century as a noun meaning "impetuosity, great speed, force, or violence (in riding, running, striking, etc.)". The word itself comes from a Middle
9807:
Although less used, the separability assumption is considered more general because every stochastic process has a separable version. It is also used when it is not possible to construct a stochastic process in a
Skorokhod space. For example, separability is assumed when constructing and studying
9193:
while posing a problem describing a random walk on the plane, which was motivated by an application in biology, but such problems involving random walks had already been studied in other fields. Certain gambling problems that were studied centuries earlier can be considered as problems involving
9119:
Other fields of probability were developed and used to study stochastic processes, with one main approach being the theory of large deviations. The theory has many applications in statistical physics, among other fields, and has core ideas going back to at least the 1930s. Later in the 1960s and
8113:
Skorokhod function spaces are frequently used in the theory of stochastic processes because it often assumed that the sample functions of continuous-time stochastic processes belong to a Skorokhod space. Such spaces contain continuous functions, which correspond to sample functions of the Wiener
1680:
Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes. Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both
1547:
random variables or random vectors in Euclidean space, so they are processes that change in discrete time. But some also use the term to refer to processes that change in continuous time, particularly the Wiener process used in financial models, which has led to some confusion, resulting in its
9977:
The term "separable" appears twice here with two different meanings, where the first meaning is from probability and the second from topology and analysis. For a stochastic process to be separable (in a probabilistic sense), its index set must be a separable space (in a topological or analytic
8833:
Probability theory has its origins in games of chance, which have a long history, with some games being played thousands of years ago, but very little analysis on them was done in terms of probability. The year 1654 is often considered the birth of probability theory when French mathematicians
549:
to study the number of phone calls occurring in a certain period of time. These two stochastic processes are considered the most important and central in the theory of stochastic processes, and were invented repeatedly and independently, both before and after Bachelier and Erlang, in different
8432:
LĂ©vy processes are types of stochastic processes that can be considered as generalizations of random walks in continuous time. These processes have many applications in fields such as finance, fluid mechanics, physics and biology. The main defining characteristics of these processes are their
9487:
for random variables associated with LĂ©vy processes. This result was later derived under more general conditions by LĂ©vy in 1934, and then Khinchin independently gave an alternative form for this characteristic function in 1937. In addition to LĂ©vy, Khinchin and Kolomogrov, early fundamental
676:
A stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set. The set used to index the random variables is called the
9525:
When constructing continuous-time stochastic processes certain mathematical difficulties arise, due to the uncountable index sets, which do not occur with discrete-time processes. One problem is that it is possible to have more than one stochastic process with the same finite-dimensional
8997:, where Kolmogorov used measure theory to develop an axiomatic framework for probability theory. The publication of this book is now widely considered to be the birth of modern probability theory, when the theories of probability and stochastic processes became parts of mathematics.
9040:
8410:
one of the main reasons for studying them. Many problems in probability have been solved by finding a martingale in the problem and studying it. Martingales will converge, given some conditions on their moments, so they are often used to derive convergence results, due largely to
8865:, which is considered a significant event in the history of probability theory. Bernoulli's book was published, also posthumously, in 1713 and inspired many mathematicians to study probability. But despite some renowned mathematicians contributing to probability theory, such as
3053:
of a stochastic process is the difference between two random variables of the same stochastic process. For a stochastic process with an index set that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period. For example, if
9198:
is based on a simple random walk, and is an example of a random walk with absorbing barriers. Pascal, Fermat and Huyens all gave numerical solutions to this problem without detailing their methods, and then more detailed solutions were presented by Jakob Bernoulli and
9060:
After World War II, the study of probability theory and stochastic processes gained more attention from mathematicians, with significant contributions made in many areas of probability and mathematics as well as the creation of new areas. Starting in the 1940s,
5006:. The index set of a stationary stochastic process is usually interpreted as time, so it can be the integers or the real line. But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.
1782:
If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process. The homogeneous Poisson process is a member of important classes of stochastic processes such as Markov processes and LĂ©vy processes.
8977:. Kolmogorov published in 1929 his first attempt at presenting a mathematical foundation, based on measure theory, for probability theory. In the early 1930s, Khinchin and Kolmogorov set up probability seminars, which were attended by researchers such as
1764:
where it is used, for example, in the BlackâScholesâMerton model. The process is also used in different fields, including the majority of natural sciences as well as some branches of social sciences, as a mathematical model for various random phenomena.
9463:'s work on Einstein's model of Brownian movement. He introduced and studied a particular set of Markov processes known as diffusion processes, where he derived a set of differential equations describing the processes. Independent of Kolmogorov's work,
8897:, where physical systems, such as containers filled with gases, are regarded or treated mathematically as collections of many moving particles. Although there were attempts to incorporate randomness into statistical physics by some scientists, such as
5030:
intuition behind stationarity is that as time passes the distribution of the stationary stochastic process remains the same. A sequence of random variables forms a stationary stochastic process only if the random variables are identically distributed.
7346:
5953:
says that there exists a modification of this process that has continuous sample paths with probability one, so the stochastic process has a continuous modification or version. The theorem can also be generalized to random fields so the index set is
9355:, but Poisson never studied the process. There are a number of claims for early uses or discoveries of the Poisson process. At the beginning of the 20th century, the Poisson process would arise independently in different situations. In Sweden 1903,
7758:
982:
is also used to refer to a stochastic or random process, though sometimes it is only used when the stochastic process takes real values. This term is also used when the index sets are mathematical spaces other than the real line, while the terms
22310:
1548:
criticism. There are various other types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines.
7338:
974:
are considered synonyms and are used interchangeably, without the index set being precisely specified. Both "collection", or "family" are used while instead of "index set", sometimes the terms "parameter set" or "parameter space" are used.
7750:
768:
When interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers, then the stochastic process is said to be in
9168:
The Bernoulli process, which can serve as a mathematical model for flipping a biased coin, is possibly the first stochastic process to have been studied. The process is a sequence of independent Bernoulli trials, which are named after
4571:
5386:
4281:
9100:, which had a strong influence on the theory of stochastic processes and stressed the importance of measure theory in probability. Doob also chiefly developed the theory of martingales, with later substantial contributions by
752:, among other terms, of a three-dimensional Wiener or Brownian motion process for time 0 †t †2. The index set of this stochastic process is the non-negative numbers, while its state space is three-dimensional Euclidean space.
19441:
Kendall, D. G.; Batchelor, G. K.; Bingham, N. H.; Hayman, W. K.; Hyland, J. M. E.; Lorentz, G. G.; Moffatt, H. K.; Parry, W.; Razborov, A. A.; Robinson, C. A.; Whittle, P. (1990). "Andrei Nikolaevich Kolmogorov (1903â1987)".
17818:
Kendall, D. G.; Batchelor, G. K.; Bingham, N. H.; Hayman, W. K.; Hyland, J. M. E.; Lorentz, G. G.; Moffatt, H. K.; Parry, W.; Razborov, A. A.; Robinson, C. A.; Whittle, P. (1990). "Andrei Nikolaevich Kolmogorov (1903â1987)".
5033:
A stochastic process with the above definition of stationarity is sometimes said to be strictly stationary, but there are other forms of stationarity. One example is when a discrete-time or continuous-time stochastic process
1705:
of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant
9512:
to prove a corresponding stochastic process exists. This theorem, which is an existence theorem for measures on infinite product spaces, says that if any finite-dimensional distributions satisfy two conditions, known as
8114:
process. But the space also has functions with discontinuities, which means that the sample functions of stochastic processes with jumps, such as the Poisson process (on the real line), are also members of this space.
9251:
wrote a paper on the method of least squares, where he used the process to study the errors of a model in time-series analysis. The work is now considered as an early discovery of the statistical method known as
9092:
in the 1950s, connecting Markov processes and potential theory, which had a significant effect on the theory of LĂ©vy processes and led to more interest in studying Markov processes with methods developed by ItĂŽ.
1756:. It can be considered as a continuous version of the simple random walk. The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled, which is the subject of
8401:
Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process (on the real line) resulting in a martingale called the
8950:. Around the start of the 20th century, mathematicians developed measure theory, a branch of mathematics for studying integrals of mathematical functions, where two of the founders were French mathematicians,
4023:
8664:
3919:
9996:
In the context of point processes, the term "state space" can mean the space on which the point process is defined such as the real line, which corresponds to the index set in stochastic process terminology.
9458:
developed in a 1931 paper a large part of the early theory of continuous-time Markov processes. Kolmogorov was partly inspired by Louis Bachelier's 1900 work on fluctuations in the stock market as well as
9443:
in 1873, preceding the work of Markov. After the work of Galton and Watson, it was later revealed that their branching process had been independently discovered and studied around three decades earlier by
6606:
9173:
who used them to study games of chance, including probability problems proposed and studied earlier by Christiaan Huygens. Bernoulli's work, including the Bernoulli process, were published in his book
8396:
6184:
6547:
7055:
6981:
6659:
668:. The theory of stochastic processes is considered to be an important contribution to mathematics and it continues to be an active topic of research for both theoretical reasons and applications.
9256:, but the work was largely overlooked. It is thought that the ideas in Thiele's paper were too advanced to have been understood by the broader mathematical and statistical community at the time.
761:
A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the
9408:
without the independence assumption, which had been commonly regarded as a requirement for such mathematical laws to hold. Markov later used Markov chains to study the distribution of vowels in
9000:
After the publication of Kolmogorov's book, further fundamental work on probability theory and stochastic processes was done by Khinchin and Kolmogorov as well as other mathematicians such as
2878:
1522:
and its value is one, while the value of a tail is zero. In other words, a Bernoulli process is a sequence of iid Bernoulli random variables, where each idealised coin flip is an example of a
8527:
3457:
5286:
22255:
4413:
2402:
There are other ways to consider a stochastic process, with the above definition being considered the traditional one. For example, a stochastic process can be interpreted or defined as a
4544:
19948:
Guttorp, Peter; Thorarinsdottir, Thordis L. (2012). "What Happened to Discrete Chaos, the Quenouille Process, and the Sharp Markov Property? Some History of Stochastic Point Processes".
10875:
Guttorp, Peter; Thorarinsdottir, Thordis L. (2012). "What Happened to Discrete Chaos, the Quenouille Process, and the Sharp Markov Property? Some History of Stochastic Point Processes".
8184:
Markov processes form an important class of stochastic processes and have applications in many areas. For example, they are the basis for a general stochastic simulation method known as
6803:
6430:
6066:
5328:
4998:
3499:
2111:
1886:
8298:
8168:
or discrete index set (often representing time), but the precise definition of a Markov chain varies. For example, it is common to define a Markov chain as a Markov process in either
6907:
6688:. The definition of separability can also be stated for other index sets and state spaces, such as in the case of random fields, where the index set as well as the state space can be
4147:
2961:
2797:
2345:
9312:
published a paper where he studied the physical observation of Brownian motion or movement to explain the seemingly random movements of particles in liquids by using ideas from the
7486:{\displaystyle \left\{X_{t}\right\},\left\{Y_{t}\right\}{\text{ uncorrelated}}\quad \iff \quad \operatorname {K} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=0\quad \forall t_{1},t_{2}}
6351:
5800:
9479:
LĂ©vy processes such as the Wiener process and the Poisson process (on the real line) are named after Paul LĂ©vy who started studying them in the 1930s, but they have connections to
5238:
relation, such as in the case of the index set being some subset of the real numbers. More formally, if a stochastic process has an index set with a total order, then a filtration
4339:
7898:{\displaystyle \left\{X_{t}\right\},\left\{Y_{t}\right\}{\text{ orthogonal}}\quad \iff \quad \operatorname {R} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=0\quad \forall t_{1},t_{2}}
7618:
7583:
7133:
7098:
3247:
3202:
3574:
3341:
5666:
5588:
4916:
2987:
2823:
2397:
1148:
914:
used the phrase "Ars Conjectandi sive Stochastice", which has been translated to "the art of conjecturing or stochastics". This phrase was used, with reference to Bernoulli, by
6855:
5926:
1102:
22300:
6472:
2615:
2215:
1042:
8800:
Other authors consider a point process as a stochastic process, where the process is indexed by sets of the underlying space on which it is defined, such as the real line or
3287:
3099:
1427:
1934:
8711:
8406:. Martingales can also be built from other martingales. For example, there are martingales based on the martingale the Wiener process, forming continuous-time martingales.
3157:
1657:, who proved its mathematical existence, but the process is also called the Brownian motion process or just Brownian motion due to its historical connection as a model for
1382:
22305:
21546:
18654:"Om Anvendelse af mindste Kvadraterbs Methode i nogle TilfĂŠlde, hvoren Komplikation af visse Slags uensartede tilfĂŠldige Fejlkilder giver Fejleneen "systematisk" Karakter"
8195:
The concept of the Markov property was originally for stochastic processes in continuous and discrete time, but the property has been adapted for other index sets such as
7990:
6686:
2730:, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take.
2143:
6498:
6397:
7142:
1181:
5444:
1215:
5686:
5412:
5158:
2066:
2046:
1954:
1906:
9728:
9675:
9622:
5866:
5537:
5204:
5098:
4823:
4753:{\displaystyle \mu _{t_{1},\dots ,t_{n}}(C)=P{\Big (}{\big \{}\omega \in \Omega :{\big (}X_{t_{1}}(\omega ),\dots ,X_{t_{n}}(\omega ){\big )}\in C{\big \}}{\Big )}.}
3766:
3646:
3616:
2661:
2371:
2243:
9505:
a suitable measurable mapping from a probability space to this measurable space of functions, and then deriving the corresponding finite-dimensional distributions.
5635:
5511:
5464:
5125:
4850:
3816:
3720:
3673:
2454:
2427:
1625:
1324:
966:
The definition of a stochastic process varies, but a stochastic process is traditionally defined as a collection of random variables indexed by some set. The terms
20308:
7627:
4433:
4091:
2272:
1744:
1724:
1297:
1264:
8553:
4779:
is a mathematical property that a stochastic process has when all the random variables of that stochastic process are identically distributed. In other words, if
1599:
1500:
18939:
9826:
9776:
9756:
9698:
9645:
9592:
9568:
9548:
9224:
8818:
8783:
8749:
8475:
8455:
8213:
8073:
8038:
8018:
7540:
7520:
6823:
6761:
6741:
6706:
6371:
6309:
6270:
6250:
6226:
6206:
6106:
6086:
6024:
6004:
5972:
5840:
5820:
5758:
5738:
5718:
5608:
5557:
5484:
5178:
5072:
5052:
5027:
4936:
4870:
4797:
4564:
4496:
4476:
4453:
4071:
3941:
3836:
3789:
3740:
3693:
3596:
3523:
3401:
3381:
3361:
3119:
3027:
3007:
2904:
2724:
2692:
2635:
2582:
2562:
2526:
2498:
2478:
2292:
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2022:
2002:
1978:
1828:
1805:
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1573:
1520:
1474:
1344:
1235:
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876:
852:
832:
719:
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7958:
7992:, and take values on the real line or on some metric space. Such functions are known as cĂ dlĂ g or cadlag functions, based on the acronym of the French phrase
21924:
9391:
which has been explained by the various applications of the process in numerous fields by biologists, ecologists, engineers and various physical scientists.
8713:, which gives the interpretation of time. Important stochastic processes such as the Wiener process, the homogeneous Poisson process (in one dimension), and
5333:
946:
French word meaning "speed, haste", and it is probably derived from a French verb meaning "to run" or "to gallop". The first written appearance of the term
22293:
21929:
4154:
9151:
The theory of stochastic processes still continues to be a focus of research, with yearly international conferences on the topic of stochastic processes.
529:
Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the
11582:
5700:
of a stochastic process is another stochastic process, which is closely related to the original stochastic process. More precisely, a stochastic process
10014:
It has been remarked that a notable exception was the St Petersburg School in Russia, where mathematicians led by Chebyshev studied probability theory.
1810:
Defined on the real line, the Poisson process can be interpreted as a stochastic process, among other random objects. But then it can be defined on the
20843:
9781:
To overcome the two difficulties described above, i.e., "more than one..." and "functionals of...", different assumptions and approaches are possible.
2500:. However this alternative definition as a "function-valued random variable" in general requires additional regularity assumptions to be well-defined.
341:
8233:
A symmetric random walk and a Wiener process (with zero drift) are both examples of martingales, respectively, in discrete and continuous time. For a
725:
is the amount that a stochastic process changes between two index values, often interpreted as two points in time. A stochastic process can have many
21539:
20667:
10541:
9924:
18873:
2663:
can represent a point in space. That said, many results and theorems are only possible for stochastic processes with a totally ordered index set.
366:
process on the surface of a sphere. The Wiener process is widely considered the most studied and central stochastic process in probability theory.
21653:
9301:
It is commonly thought that Bachelier's work gained little attention and was forgotten for decades until it was rediscovered in the 1950s by the
9235:
the lattice an infinite number of times with probability one in one and two dimensions, but with probability zero in three or higher dimensions.
9136:
were introduced and developed to study stochastic processes and other mathematical objects in probability theory, which respectively resulted in
2746:
of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process. More precisely, if
9324:, for describing the probability of finding a particle in a certain region of space. Shortly after Einstein's first paper on Brownian movement,
22288:
21270:
20263:
8820:-dimensional Euclidean space. Other stochastic processes such as renewal and counting processes are studied in the theory of point processes.
8398:
is a discrete-time martingale. In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables.
20800:
20780:
7928:, is a mathematical space of all the functions that are right-continuous with left limits, defined on some interval of the real line such as
9363:
containing work, now considered fundamental and pioneering, where he proposed to model insurance claims with a homogeneous Poisson process.
9047:
did early work on the theory of stochastic processes, making fundamental contributions, particularly in the theory of martingales. His book
22106:
9484:
19985:
Jarrow, Robert; Protter, Philip (2004). "A short history of stochastic integration and mathematical finance: the early years, 1880â1970".
18901:"Bachelier: Not the forgotten forerunner he has been depicted as. An analysis of the dissemination of Louis Bachelier's work in economics"
10724:
Jarrow, Robert; Protter, Philip (2004). "A short history of stochastic integration and mathematical finance: the early years, 1880â1970".
10050:
The theorem has other names including Kolmogorov's consistency theorem, Kolmogorov's extension theorem or the DaniellâKolmogorov theorem.
9266:
gave the first mathematical proof of the existence of the Wiener process. This mathematical object had appeared previously in the work of
9128:
in the United States of America, which would later result in Varadhan winning the 2007 Abel Prize. In the 1990s and 2000s the theories of
21532:
21184:
18784:
17105:
David, F. N. (1955). "Studies in the History of Probability and Statistics I. Dicing and Gaming (A Note on the History of Probability)".
17420:
Bingham, N. (2000). "Studies in the history of probability and statistics XLVI. Measure into probability: from Lebesgue to Kolmogorov".
15387:
Kun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3
9328:
published work where he cited Einstein, but wrote that he had independently derived the equivalent results by using a different method.
9080:
Also starting in the 1940s, connections were made between stochastic processes, particularly martingales, and the mathematical field of
3948:
1502:. This process can be linked to an idealisation of repeatedly flipping a coin, where the probability of obtaining a head is taken to be
22151:
8560:
8153:
The Brownian motion process and the Poisson process (in one dimension) are both examples of Markov processes in continuous time, while
3843:
557:
is also used to refer to a stochastic or random process, because a stochastic process can also be interpreted as a random element in a
17:
9508:
Another approach involves defining a collection of random variables to have specific finite-dimensional distributions, and then using
9247:
or Brownian motion process has its origins in different fields including statistics, finance and physics. In 1880, Danish astronomer
21101:
18802:
8985:, and Khinchin gave the first mathematical definition of a stochastic process as a set of random variables indexed by the real line.
8040:. The notation of this function space can also include the interval on which all the cĂ dlĂ g functions are defined, so, for example,
21658:
20785:
19852:
8931:
9375:
He then found the limiting case, which is effectively recasting the Poisson distribution as a limit of the binomial distribution.
6552:
5930:
holds. Two stochastic processes that are modifications of each other have the same finite-dimensional law and they are said to be
4762:
The finite-dimensional distributions of a stochastic process satisfy two mathematical conditions known as consistency conditions.
22067:
21111:
20795:
8675:, but the processes are often defined so that they take values in Euclidean space. The index set is the non-negative numbers, so
8238:
1453:
8303:
21153:
21050:
6117:
573:
for the set that indexes the random variables. But often these two terms are used when the random variables are indexed by the
334:
6503:
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20197:
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13141:
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12998:
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12876:
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11793:
11766:
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11698:
11630:
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11539:
11486:
11459:
11432:
11405:
11378:
11348:
11321:
11294:
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11240:
11213:
11186:
11159:
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11090:
11056:
10998:
10960:
10933:
10852:
10751:
10672:
10645:
10614:
10580:
10549:
10518:
10487:
10426:
10395:
10364:
10333:
10302:
10271:
10241:
10191:
10155:
10123:
9298:, but Bachelier did not cite him, and Bachelier's thesis is now considered pioneering in the field of financial mathematics.
6986:
6912:
6611:
22234:
21240:
21204:
17458:
Benzi, Margherita; Benzi, Michele; Seneta, Eugene (2007). "Francesco Paolo Cantelli. b. 20 December 1875 d. 21 July 1966".
8885:, most of the mathematical community did not consider probability theory to be part of mathematics until the 20th century.
21688:
21683:
21508:
21245:
10032:
Doob, when citing Khinchin, uses the term 'chance variable', which used to be an alternative term for 'random variable'.
9838:
5234:
is an increasing sequence of sigma-algebras defined in relation to some probability space and an index set that has some
5486:. With the concept of a filtration, it is possible to study the amount of information contained in a stochastic process
21157:
20355:
20256:
19086:
17907:
17871:
17198:
12914:
2833:
787:
9452:
became interested in Markov chains, eventually resulting in him publishing in 1938 a detailed study on Markov chains.
8965:
In the 1920s, fundamental contributions to probability theory were made in the Soviet Union by mathematicians such as
8480:
3423:
21310:
19880:
18900:
18287:
Ellis, Richard S. (1995). "An overview of the theory of large deviations and applications to statistical mechanics".
10456:
9480:
9294:, without knowing the work of Thiele. It has been speculated that Bachelier drew ideas from the random walk model of
8169:
8147:
5241:
693:, giving the index set the interpretation of time. Each random variable in the collection takes values from the same
597:
instead. The values of a stochastic process are not always numbers and can be vectors or other mathematical objects.
327:
315:
274:
8188:, which is used for simulating random objects with specific probability distributions, and has found application in
4344:
22101:
21785:
21355:
21161:
21145:
21060:
20888:
20858:
20280:
10637:
10606:
10572:
10387:
600:
Based on their mathematical properties, stochastic processes can be grouped into various categories, which include
18833:
Courtault, Jean-Michel; Kabanov, Yuri; Bru, Bernard; Crepel, Pierre; Lebon, Isabelle; Le Marchand, Arnaud (2000).
8668:
are all independent of each other, and the distribution of each increment only depends on the difference in time.
4504:
22369:
21848:
21608:
21260:
21225:
21194:
21189:
20625:
20542:
18769:
18717:
Lauritzen, Steffen L. (1981). "Time Series Analysis in 1880: A Discussion of Contributions Made by T.N. Thiele".
9853:
9468:
8982:
8224:
5231:
4048:
1753:
605:
205:
141:
9987:
The definition of separability for a continuous-time real-valued stochastic process can be stated in other ways.
9808:
random fields, where the collection of random variables is now indexed by sets other than the real line such as
9427:
with an aim to study card shuffling. Other early uses of Markov chains include a diffusion model, introduced by
6766:
6402:
6029:
5949:
If a continuous-time real-valued stochastic process meets certain moment conditions on its increments, then the
5291:
4943:
3462:
2074:
1849:
1456:(iid) random variables, where each random variable takes either the value one or zero, say one with probability
21775:
21199:
20828:
20823:
20630:
20527:
9464:
8411:
8173:
it has discrete time, now tends to be used, despite the second definition having been used by researchers like
1681:
continuous index set and states space. But the process can be defined more generally so its state space can be
253:
114:
9028:
greatly interrupted the development of probability theory, causing, for example, the migration of Feller from
8244:
8215:-dimensional Euclidean space, which results in collections of random variables known as Markov random fields.
6860:
22359:
21780:
21673:
21513:
21290:
21126:
21025:
21010:
20549:
20422:
20338:
20249:
18013:
Getoor, Ronald (2009). "J. L. Doob: Foundations of stochastic processes and probabilistic potential theory".
9884:
9129:
8765:
A point process is a collection of points randomly located on some mathematical space such as the real line,
5950:
4100:
2913:
2749:
2297:
21285:
21165:
9445:
6314:
5763:
22334:
21678:
21295:
9509:
4288:
21300:
20936:
20142:
19069:
Embrechts, Paul; Frey, RĂŒdiger; Furrer, Hansjörg (2001). "Stochastic processes in insurance and finance".
9109:
9013:
8959:
7588:
7553:
7103:
7068:
3207:
3162:
898:
was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a
21720:
20898:
20482:
20427:
20343:
9874:
9024:. Decades later, Cramér referred to the 1930s as the "heroic period of mathematical probability theory".
8993:
In 1933, Andrei Kolmogorov published in German, his book on the foundations of probability theory titled
3534:
3292:
1601:, so the index set of this random walk is the natural numbers, while its state space is the integers. If
21230:
9952:, and the stochastic process, a mathematical object, but to avoid ambiguity this article uses the terms
5640:
5562:
4875:
3363:-valued random variable known as an increment. When interested in the increments, often the state space
2966:
2802:
2376:
1107:
22364:
22338:
21802:
21235:
21220:
20863:
20833:
20400:
20298:
18834:
15712:
Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues
15407:
Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues
14456:
Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues
8714:
8671:
A LĂ©vy process can be defined such that its state space is some abstract mathematical space, such as a
6832:
5873:
1267:
1059:
109:
6435:
2591:
2170:
1018:
729:, due to its randomness, and a single outcome of a stochastic process is called, among other names, a
22198:
22131:
22126:
22089:
21620:
21593:
21563:
21315:
21116:
21030:
21015:
20946:
20522:
20405:
20303:
19862:
12252:
12012:
9405:
3252:
3057:
1387:
930:. For the term and a specific mathematical definition, Doob cited another 1934 paper, where the term
903:
225:
1915:
22055:
22050:
21959:
21919:
21635:
21149:
21035:
20537:
20512:
20457:
16932:
10734:
8678:
8185:
6229:
5003:
3124:
1349:
284:
279:
168:
153:
8946:, where his sixth problem asked for a mathematical treatment of physics and probability involving
7333:{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=\operatorname {E} \left}
455:
of systems and phenomena that appear to vary in a random manner. Examples include the growth of a
22244:
21450:
21440:
21255:
21131:
20913:
20838:
20652:
20517:
20373:
20328:
12257:
12017:
11082:
Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, Via Conditioning
10356:
10325:
9313:
7963:
6664:
2116:
263:
134:
21880:
9789:
One approach for avoiding mathematical construction issues of stochastic processes, proposed by
6477:
6376:
22249:
22239:
22213:
22035:
22030:
22025:
22020:
21978:
21941:
21887:
21613:
21603:
21581:
21392:
21320:
20745:
20735:
20579:
10729:
10041:
Later translated into English and published in 1950 as Foundations of the Theory of Probability
9074:
2457:
1646:
1153:
798:
If the state space is the integers or natural numbers, then the stochastic process is called a
578:
158:
20041:
19510:
18761:
18342:
14850:
14630:
14535:
13961:
12415:
12280:
9339:, and Fourier analysis to prove the existence of the Wiener process as a mathematical object.
5417:
1185:
22166:
22146:
22084:
22013:
21792:
21770:
21745:
21645:
21415:
21397:
21377:
21372:
21091:
20923:
20903:
20750:
20693:
20532:
20442:
19810:
18599:
18518:
18491:
18220:
16654:
16347:
15818:
14964:
14707:
14307:
14122:
14066:
13158:
12839:
12731:
12674:
12644:
12265:
12202:
Kolmogoroff, A. (1931). "Ăber die analytischen Methoden in der Wahrscheinlichkeitsrechnung".
12025:
11726:
9417:
9317:
8943:
8894:
7745:{\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }(t_{1},t_{2})=\operatorname {E} }
5671:
5391:
5130:
2743:
2051:
2031:
1939:
1891:
1642:
991:
are usually used when the index set is interpreted as time, and other terms are used such as
954:, which the Oxford English Dictionary also gives as a synonym, and was used in an article by
915:
726:
649:
299:
258:
163:
129:
20163:
19415:
18193:
18166:
17501:
Doob, Joseph L. (1996). "The Development of Rigor in Mathematical Probability (1900-1950)".
17310:
17149:
17052:
16946:
16688:
16627:
16512:
16485:
16431:
16374:
16313:
16283:
16239:
15710:
15683:
15405:
15201:
15171:
15144:
14734:
14596:
14454:
14153:
13928:
13672:
13572:
13491:
13410:
13383:
13326:
13215:
13131:
12514:
12460:
12314:
12067:
11875:
11688:
11586:
11556:
11449:
11284:
11203:
11080:
10145:
10113:
9707:
9654:
9601:
8846:. But there was earlier mathematical work done on the probability of gambling games such as
5845:
5516:
5183:
5077:
4802:
3745:
3621:
3601:
2640:
2350:
2222:
22324:
22218:
22203:
22118:
22045:
22008:
22003:
21993:
21753:
21707:
21490:
21445:
21435:
21121:
21096:
21065:
21045:
20883:
20805:
20790:
20657:
19900:
Applebaum, David (2004). "LĂ©vy processes: From probability to finance and quantum groups".
19380:
18344:
Conformal Invariance: an Introduction to Loops, Interfaces and Stochastic Loewner Evolution
18032:
16719:
Applebaum, David (2004). "LĂ©vy processes: From probability to finance and quantum groups".
16584:
13598:
Applebaum, David (2004). "LĂ©vy processes: From probability to finance and quantum groups".
13111:
Applebaum, David (2004). "LĂ©vy processes: From probability to finance and quantum groups".
12113:
11665:
Applebaum, David (2004). "LĂ©vy processes: From probability to finance and quantum groups".
11502:
Applebaum, David (2004). "LĂ©vy processes: From probability to finance and quantum groups".
9848:
9440:
9352:
9335:, later inspired Norbert Wiener in the 1920s to use a type of measure theory, developed by
8866:
5613:
5489:
5449:
5103:
4828:
3794:
3698:
3651:
2432:
2405:
1757:
1650:
1604:
1302:
938:, though the German term had been used earlier, for example, by Andrei Kolmogorov in 1931.
289:
183:
76:
16845:
16818:
14655:
14007:
12933:
Probability Theory and Statistical Inference: Econometric Modeling with Observational Data
10925:
Theory of Stochastic Processes: With Applications to Financial Mathematics and Risk Theory
4418:
4076:
2248:
1729:
1709:
1273:
1240:
8:
22193:
22062:
21973:
21860:
21831:
21555:
21485:
21325:
21250:
21055:
20815:
20725:
20615:
19989:. Institute of Mathematical Statistics Lecture Notes - Monograph Series. pp. 75â91.
19045:
An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods
18409:
An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods
17081:
An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods
16903:
An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods
13841:
An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods
13760:
An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods
11397:
An Introduction to the Theory of Point Processes: Volume II: General Theory and Structure
10728:. Institute of Mathematical Statistics Lecture Notes - Monograph Series. pp. 75â80.
9909:
9869:
9843:
9701:
9595:
9325:
9066:
8902:
8532:
8189:
3526:
1981:
1578:
1479:
665:
424:
248:
190:
178:
173:
19384:
19241:
Seneta, E. (1998). "I.J. Bienaymé : Criticality, Inequality, and Internationalization".
18036:
16588:
12117:
9550:, other characteristics that depend on an uncountable number of points of the index set
9101:
1843:
A stochastic process is defined as a collection of random variables defined on a common
22330:
21983:
21951:
21934:
21914:
21897:
21807:
21630:
21455:
21420:
21335:
21305:
21136:
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21070:
20893:
20730:
20395:
20333:
19973:
19266:
19024:
18981:
18931:
18865:
18742:
18696:
18056:
18022:
17793:
17715:
17672:
17629:
17526:
17483:
17391:
17130:
16608:
16574:
14202:
13093:
12227:
12184:
12136:
12101:
11972:
10900:
10823:
10815:
9965:
9914:
9899:
9811:
9797:
9761:
9741:
9683:
9630:
9577:
9553:
9533:
9321:
9209:
9125:
8970:
8803:
8768:
8734:
8460:
8440:
8198:
8043:
8023:
8003:
7997:
7525:
7505:
6808:
6746:
6726:
6691:
6356:
6294:
6255:
6235:
6211:
6191:
6091:
6071:
6009:
5989:
5957:
5825:
5805:
5743:
5723:
5703:
5593:
5542:
5469:
5207:
5163:
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5012:
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4771:
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3366:
3346:
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3012:
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2672:
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2511:
2483:
2463:
2277:
2148:
2007:
1987:
1963:
1813:
1790:
1684:
1558:
1505:
1459:
1329:
1220:
998:
935:
861:
837:
817:
806:. If the state space is the real line, then the stochastic process is referred to as a
704:
694:
570:
515:
507:
503:
460:
452:
371:
124:
64:
41:
20212:
19078:
16096:
Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Second Edition
13954:
12408:
12159:
Khintchine, A. (1934). "Korrelationstheorie der stationeren stochastischen Prozesse".
12006:
11558:
Upper and Lower Bounds for Stochastic Processes: Modern Methods and Classical Problems
10690:
9517:, then there exists a stochastic process with those finite-dimensional distributions.
9449:
9009:
8081:
7931:
5381:{\displaystyle {\mathcal {F}}_{s}\subseteq {\mathcal {F}}_{t}\subseteq {\mathcal {F}}}
1726:, which is a real number, then the resulting stochastic process is said to have drift
22156:
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19396:
19351:
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19297:
19258:
19191:
19161:
19127:
19110:
Cramér, Harald (1969). "Historical review of Filip Lundberg's works on risk theory".
19092:
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10267:
10237:
10187:
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10119:
9919:
9904:
9801:
9455:
9432:
9413:
9386:
published experimental results on counting alpha particles. Motivated by their work,
9379:
9200:
9121:
9088:
and then later work by Joseph Doob. Further work, considered pioneering, was done by
8974:
8919:
8893:
In the physical sciences, scientists developed in the 19th century the discipline of
8882:
8870:
8843:
8158:
4276:{\displaystyle \mu _{t_{1},\dots ,t_{n}}=P\circ (X({t_{1}}),\dots ,X({t_{n}}))^{-1},}
2826:
1844:
1658:
1449:
1443:
955:
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436:
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200:
99:
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20931:
19157:
18869:
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17863:
17190:
16612:
13921:
12906:
10023:
The name Khinchin is also written in (or transliterated into) English as Khintchine.
9348:
8878:
628:. The study of stochastic processes uses mathematical knowledge and techniques from
22074:
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21853:
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17254:
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15169:
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13077:
12902:
12211:
12168:
12131:
12121:
10884:
10799:
10739:
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10229:
9648:
9489:
9253:
9105:
9081:
9017:
8966:
8907:
8851:
895:
661:
617:
519:
511:
495:
381:
119:
49:
20487:
17733:
Brush, S.G. (1958). "The development of the kinetic theory of gases IV. Maxwell".
12246:
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10923:
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10662:
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10535:
10504:
10473:
10412:
10381:
10350:
10319:
10288:
10261:
10231:
10181:
10089:
9864:
9309:
9283:
9275:
9271:
9267:
9248:
9227:
9170:
9141:
9085:
8915:
8898:
8862:
8858:
8793:
denoted an evolution of some system in time, so a point process is also called a
7915:
6285:
4499:
2727:
2541:
1773:
1523:
911:
777:
690:
621:
590:
586:
542:
534:
432:
363:
195:
146:
21275:
20507:
18653:
17054:
Stochastic Geometry, Spatial Statistics and Random Fields: Models and Algorithms
16629:
Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences
9286:
used a Wiener process in his 1900 thesis in order to model price changes on the
9231:
8427:
613:
21824:
21465:
21430:
21350:
20956:
20703:
20620:
20589:
20584:
20564:
20554:
20497:
20472:
20452:
20417:
20385:
20368:
20183:
20161:
20103:
16596:
14594:
12106:
Proceedings of the National Academy of Sciences of the United States of America
10919:
10257:
9460:
9436:
9428:
9356:
9302:
9295:
9291:
9263:
9244:
9120:
1970s, fundamental work was done by Alexander Wentzell in the Soviet Union and
9005:
8978:
8951:
8300:
with zero mean, the stochastic process formed from the successive partial sums
1654:
1636:
899:
657:
637:
609:
558:
530:
428:
359:
210:
20492:
19994:
18322:
Raussen, Martin; Skau, Christian (2008). "Interview with Srinivasa Varadhan".
17746:
17433:
10743:
8955:
8122:
In the context of mathematical construction of stochastic processes, the term
775:. If the index set is some interval of the real line, then time is said to be
22353:
22141:
22096:
22079:
21725:
21598:
21367:
20908:
20740:
20698:
20640:
20462:
20378:
20318:
20012:
19969:
19940:
19931:
19914:
19463:
19400:
19262:
19131:
19096:
19020:
18977:
18927:
18861:
18738:
18692:
18308:
18273:
18264:
18247:
18104:
18095:
18078:
18052:
17994:
17985:
17968:
17953:
17944:
17927:
17840:
17789:
17754:
17711:
17668:
17625:
17590:
17581:
17564:
17522:
17479:
17441:
17387:
17268:
17259:
17242:
17178:
17126:
16936:
16604:
14402:
Multidimensional Stochastic Processes as Rough Paths: Theory and Applications
14197:
14180:
13089:
12223:
12180:
11525:
11478:
Functional Analysis for Probability and Stochastic Processes: An Introduction
10896:
10811:
10761:
9961:
9409:
9400:
9387:
9336:
9145:
9033:
8939:
8839:
8835:
8760:
8178:
8076:
5975:
1957:
1760:
or invariance principle, also known as the functional central limit theorem.
1749:
771:
653:
464:
83:
18853:
17647:
Brush, Stephen G. (1967). "Foundations of statistical mechanics 1845?1915".
9493:
9062:
9036:
and the death of Doeblin, considered now a pioneer in stochastic processes.
1668:
Realizations of Wiener processes (or Brownian motion processes) with drift (
22161:
22040:
21819:
21730:
21625:
21425:
21387:
20941:
20873:
20762:
20757:
20569:
20502:
20477:
20313:
20020:
Meyer, Paul-André (2009). "Stochastic Processes from 1950 to the Present".
19455:
18127:
Meyer, Paul-André (2009). "Stochastic Processes from 1950 to the Present".
17832:
13241:
12145:
9889:
9879:
9858:
9424:
9287:
9186:
9137:
9089:
9025:
8911:
8726:
8672:
8141:
3404:
1909:
1540:
906:
gives the year 1662 as its earliest occurrence. In his work on probability
594:
546:
538:
487:
310:
220:
104:
20120:
A History of Probability and Statistics and Their Applications before 1750
19151:
19149:
18574:
A History of Probability and Statistics and Their Applications before 1750
18547:
A History of Probability and Statistics and Their Applications before 1750
18436:
A History of Probability and Statistics and Their Applications before 1750
17285:
A History of Probability and Statistics and Their Applications before 1750
14151:
12126:
8962:
published the first probability book that used ideas from measure theory.
8901:, most of the work had little or no randomness. This changed in 1859 when
8785:-dimensional Euclidean space, or more abstract spaces. Sometimes the term
941:
According to the Oxford English Dictionary, early occurrences of the word
22272:
22188:
21470:
21005:
20989:
20984:
20979:
20969:
20772:
20713:
20708:
20672:
20432:
20323:
16846:
Sung Nok Chiu; Dietrich Stoyan; Wilfrid S. Kendall; Joseph Mecke (2013).
16819:
Sung Nok Chiu; Dietrich Stoyan; Wilfrid S. Kendall; Joseph Mecke (2013).
16565:
Glasserman, Paul; Kou, Steven (2006). "A Conversation with Chris Heyde".
16343:
14656:
Sung Nok Chiu; Dietrich Stoyan; Wilfrid S. Kendall; Joseph Mecke (2013).
14008:
Sung Nok Chiu; Dietrich Stoyan; Wilfrid S. Kendall; Joseph Mecke (2013).
9790:
9383:
9371:
9332:
9133:
9044:
9001:
8174:
8165:
8154:
6712:
5235:
2585:
1664:
1535:
927:
762:
629:
601:
230:
71:
59:
19320:
Paul Lévy and Maurice Fréchet: 50 Years of Correspondence in 107 Letters
19316:
19227:
16996:
10790:
Stirzaker, David (2000). "Advice to Hedgehogs, or, Constants Can Vary".
8854:, written in the 16th century but posthumously published later in 1663.
4027:
The law of a stochastic process or a random variable is also called the
781:. The two types of stochastic processes are respectively referred to as
22208:
21480:
21020:
20964:
20848:
20801:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
20241:
19270:
19012:
18969:
18815:
18779:
18746:
18700:
17781:
17703:
17660:
17617:
17530:
17395:
17134:
14396:
13081:
12215:
12172:
11523:
10819:
9894:
9530:
well-defined random variable. For a continuous-time stochastic process
9113:
8874:
2025:
1653:
based on the size of the increments. The Wiener process is named after
701:. This state space can be, for example, the integers, the real line or
641:
448:
88:
34:
19392:
19243:
International Statistical Review / Revue Internationale de Statistique
18719:
International Statistical Review / Revue Internationale de Statistique
18673:
International Statistical Review / Revue Internationale de Statistique
18044:
12432:
Probability and Random Processes for Electrical and Computer Engineers
9160:
processes have long and complex histories, stretching back centuries.
6711:
The concept of separability of a stochastic process was introduced by
834:-dimensional Euclidean space, then the stochastic process is called a
22256:
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
21571:
20974:
19915:"Half a Century with Probability Theory: Some Personal Recollections"
19754:
19697:
19535:
17565:"Half a Century with Probability Theory: Some Personal Recollections"
16579:
15097:
10510:
9070:
9051:
is considered highly influential in the field of probability theory.
5974:-dimensional Euclidean space as well as to stochastic processes with
2703:
2564:
the interpretation of time. In addition to these sets, the index set
2537:
686:
678:
582:
479:
474:. Stochastic processes have applications in many disciplines such as
440:
19566:
Stochastic Differential Equations: An Introduction with Applications
19254:
18730:
18684:
17514:
17379:
17118:
15604:
Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives
14936:
Stochastic Differential Equations: An Introduction with Applications
14852:
Diffusions, Markov Processes, and Martingales: Volume 1, Foundations
14537:
Diffusions, Markov Processes, and Martingales: Volume 1, Foundations
14363:
Stochastic Differential Equations: An Introduction with Applications
12316:
Diffusions, Markov Processes, and Martingales: Volume 1, Foundations
12042:
Theory of probability and statistics as exemplified in short dictums
11690:
Diffusions, Markov Processes, and Martingales: Volume 1, Foundations
11205:
Diffusions, Markov Processes, and Martingales: Volume 1, Foundations
10803:
10446:
10115:
Diffusions, Markov Processes, and Martingales: Volume 1, Foundations
9259:
8164:
A Markov chain is a type of Markov process that has either discrete
3029:. Other names for a sample function of a stochastic process include
740:
18671:
Hald, Anders (1981). "T. N. Thiele's Contributions to Statistics".
15102:
10294:
8234:
1045:
645:
633:
574:
471:
456:
215:
18027:
16065:
4018:{\displaystyle \mu (B)=P(\{\omega \in \Omega :X(\omega )\in B\}).}
3648:
is the pre-image of the measurable function or, equivalently, the
20060:
19727:
18387:
Quastel, Jeremy (2015). "The Work of the 2014 Fields Medalists".
16146:
15600:
14251:
13297:
12698:
12696:
11781:
11120:
10840:
9367:
8659:{\displaystyle X_{t_{2}}-X_{t_{1}},\dots ,X_{t_{n}}-X_{t_{n-1}},}
6291:
More precisely, a real-valued continuous-time stochastic process
3914:{\displaystyle X^{-1}(B)=\{\omega \in \Omega :X(\omega )\in B\},}
2699:
2219:
Historically, in many problems from the natural sciences a point
1752:, a sample path of a Wiener process is continuous everywhere but
523:
491:
483:
475:
19417:
Continuous-Time Markov Chains: An Applications-Oriented Approach
18809:. Serie 3, 17: 21â89 (Engl. translation by David R. May, 2011).
17000:
Statistical Inference and Simulation for Spatial Point Processes
16537:
13013:
12278:
8731:
A random field is a collection of random variables indexed by a
8433:
stationarity and independence properties, so they were known as
2536:
of the stochastic process. Often this set is some subset of the
22311:
European Community on Computational Methods in Applied Sciences
21875:
21693:
20230:
20085:
Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues
19187:
Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues
18835:"Louis Bachelier on the Centenary of Theorie de la Speculation"
16625:
16177:
Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues
14395:
9483:
going back to the 1920s. In a 1932 paper, Kolmogorov derived a
9360:
9154:
9029:
6288:, which means that the index set has a dense countable subset.
4546:, the finite-dimensional distributions of a stochastic process
4042:
3768:, so the law of a stochastic process is a probability measure.
682:
18466:
Nonequilibrium phenomena II: from stochastics to hydrodynamics
17813:
17811:
17809:
17807:
16092:
15685:
An Introduction to Stochastic Processes and Their Applications
15203:
An Introduction to Stochastic Processes and Their Applications
15170:
Iosif Ilyich Gikhman; Anatoly Vladimirovich Skorokhod (1969).
14733:
Iosif Ilyich Gikhman; Anatoly Vladimirovich Skorokhod (1969).
14003:
14001:
13130:
Iosif Ilyich Gikhman; Anatoly Vladimirovich Skorokhod (1969).
12693:
12065:
11939:
11370:
Introduction to Matrix Analytic Methods in Stochastic Modeling
10661:
Iosif Ilyich Gikhman; Anatoly Vladimirovich Skorokhod (1969).
10230:
Iosif Ilyich Gikhman; Anatoly Vladimirovich Skorokhod (1969).
9039:
593:, then the collection of random variables is usually called a
19440:
19145:
19143:
19141:
17890:
Vere-Jones, David (2006). "Khinchin, Aleksandr Yakovlevich".
17854:
Vere-Jones, David (2006). "Khinchin, Aleksandr Yakovlevich".
17817:
17023:
16990:
16531:
16237:
16038:
15930:
15846:
15648:
14848:
14844:
14842:
14533:
13891:
13670:
13616:
13570:
13489:
13408:
13381:
13213:
13186:
12312:
11754:
11686:
11682:
11680:
11618:
11309:
11201:
10870:
10868:
10866:
10864:
10111:
9331:
Einstein's work, as well as experimental results obtained by
8947:
8935:
5054:
is said to be stationary in the wide sense, then the process
3403:-dimensional Euclidean space or more abstract spaces such as
1543:
are stochastic processes that are usually defined as sums of
1346:, and not the entire stochastic process. If the index set is
354:
22306:
International Council for Industrial and Applied Mathematics
20022:
Electronic Journal for History of Probability and Statistics
19637:
18129:
Electronic Journal for History of Probability and Statistics
16764:
16233:
16231:
16229:
16059:
15163:
14155:
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13469:. Springer Science & Business Media. pp. 225, 260.
13456:
12783:
12308:
12306:
12304:
12302:
11147:
10471:
10225:
10223:
8842:
had a written correspondence on probability, motivated by a
8146:
Markov processes are stochastic processes, traditionally in
6601:{\displaystyle \{X_{t}\in F{\text{ for all }}t\in G\cap U\}}
2347:
to reflect that it is actually a function of two variables,
522:
have motivated the extensive use of stochastic processes in
19677:. Springer Science & Business Media. pp. 529â530.
19434:
18999:
Brush, Stephen G. (1968). "A history of random processes".
18956:
Brush, Stephen G. (1968). "A history of random processes".
17804:
17768:
Brush, Stephen G. (1968). "A history of random processes".
16893:
16011:
15661:. Springer Science & Business Media. pp. 148â165.
13998:
13783:
13207:
13068:
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13007:
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10418:
10318:
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10221:
10219:
10217:
10215:
10213:
10211:
10209:
10207:
10205:
10203:
10083:
10081:
10079:
10077:
10075:
10073:
10071:
10069:
10067:
9488:
contributions to the theory of LĂ©vy processes were made by
8161:
problem are examples of Markov processes in discrete time.
4852:
has the same distribution, which means that for any set of
2274:
is a random variable representing a value observed at time
1702:
444:
390:
20781:
Autoregressive conditional heteroskedasticity (ARCH) model
20162:
Iosif I. Gikhman; Anatoly Vladimirovich Skorokhod (1996).
19138:
18340:
18334:
16318:. Springer Science & Business Media. pp. 12, 13.
15842:
15840:
15658:
Multiparameter Processes: An Introduction to Random Fields
15542:
15540:
15538:
15448:
15446:
15444:
15442:
15346:
Multiparameter Processes: An Introduction to Random Fields
15252:
15250:
14839:
14726:
14595:
Iosif I. Gikhman; Anatoly Vladimirovich Skorokhod (1969).
14588:
14527:
14391:
14389:
14387:
14385:
14383:
14231:. Springer Science & Business Media. pp. 93, 94.
14145:
13973:
13971:
13831:
13439:
Stochastic Calculus for Finance II: Continuous-Time Models
13182:
13180:
13123:
12059:
11978:
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11967:
11965:
11963:
11935:
11933:
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11677:
11393:
10861:
10633:
Stochastic Calculus for Finance II: Continuous-Time Models
10321:
Stochastic Population Dynamics in Ecology and Conservation
9055:
8391:{\displaystyle X_{1},X_{1}+X_{2},X_{1}+X_{2}+X_{3},\dots }
1056:
A stochastic process can be denoted, among other ways, by
19748:
19723:
19721:
19691:
19062:
19035:
18832:
18399:
18158:
17690:
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17604:
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16714:
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16710:
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Stochastic processes: a survey of the mathematical theory
15594:
15429:
15427:
15410:. Springer Science & Business Media. pp. 78â79.
15206:. Springer Science & Business Media. pp. 19â20.
14996:. Springer Science & Business Media. pp. 18â19.
14571:
Stochastic processes: a survey of the mathematical theory
14432:. Springer Science & Business Media. pp. 40â41.
13763:. Springer Science & Business Media. pp. 19â36.
13750:
13664:
13483:
13402:
13375:
13350:
13348:
12589:
Stochastic processes: a survey of the mathematical theory
12299:
11807:
11805:
11750:
11748:
11614:
11612:
11610:
11608:
11048:
Stochastic processes: a survey of the mathematical theory
10993:. Springer Science & Business Media. pp. 24â25.
10107:
10105:
10103:
10101:
8418:
calculus and other fields such as economics and finance.
6179:{\displaystyle P(X_{t}=Y_{t}{\text{ for all }}t\in T)=1,}
2989:, there exists a sample function that maps the index set
1544:
468:
408:
399:
20309:
Independent and identically distributed random variables
19812:
Extremes in Random Fields: A Theory and Its Applications
19470:
19214:
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18828:
18826:
18712:
18710:
16785:
16758:
16731:
15336:
15010:
14818:
13885:
13610:
13516:
13510:
13354:
11916:. World Scientific Publishing Co Inc. pp. 177â178.
11660:
11658:
11656:
11654:
11652:
11650:
11648:
11646:
11644:
11642:
11362:
11360:
10527:
10200:
10064:
8925:
7497:
6542:{\displaystyle F\subset \textstyle R=(-\infty ,\infty )}
1326:
are used to refer to the random variable with the index
19347:
Basic Principles and Applications of Probability Theory
19048:. Springer Science & Business Media. pp. 8â9.
19041:
18908:
The European Journal of the History of Economic Thought
18412:. Springer Science & Business Media. pp. 1â4.
18405:
17885:
17883:
17077:
17017:
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16682:
16680:
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15247:
15195:
15193:
15091:
14958:
14956:
14812:
14701:
14699:
14473:
14380:
14326:
14282:
Basic Principles and Applications of Probability Theory
14228:
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14220:
14218:
14216:
14152:
Georg Lindgren; Holger Rootzen; Maria Sandsten (2013).
14117:
14115:
14068:
Poisson Point Processes: Imaging, Tracking, and Sensing
13968:
13837:
13756:
13564:
13177:
12542:
12538:
12536:
12272:
11960:
11943:
Numerical Solution of Stochastic Differential Equations
11930:
11905:
11903:
11901:
11899:
11897:
11366:
11127:. Springer Science & Business Media. pp. 7â8.
11116:
11114:
11112:
11110:
11108:
11106:
11104:
11102:
10982:
10980:
10978:
10976:
10974:
10972:
10952:
Basic Principles and Applications of Probability Theory
10917:
10594:
10592:
3383:
is the real line or the natural numbers, but it can be
21554:
20786:
Autoregressive integrated moving average (ARIMA) model
19947:
19788:. Springer Science & Business Media. p. 536.
19761:. Springer Science & Business Media. p. 211.
19718:
19704:. Springer Science & Business Media. p. 221.
19420:. Springer Science & Business Media. p. vii.
19350:. Springer Science & Business Media. p. 146.
19310:
18347:. Springer Science & Business Media. p. 113.
18198:. Springer Science & Business Media. p. 176.
17339:
Probability and Statistics: The Science of Uncertainty
17217:
Probability and Statistics: The Science of Uncertainty
16906:. Springer Science & Business Media. p. 194.
16839:
16812:
16771:. Springer Science & Business Media. p. 171.
16707:
16490:. Springer Science & Business Media. p. 371.
16379:. Springer Science & Business Media. p. 115.
16337:
16335:
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16180:. Springer Science & Business Media. p. 253.
15978:
15783:
15634:. Springer Science & Business Media. p. 532.
15607:. Springer Science & Business Media. p. 154.
15521:. Springer Science & Business Media. p. 420.
15424:
15349:. Springer Science & Business Media. p. 155.
15307:
15293:. Springer Science & Business Media. p. 535.
15233:. Springer Science & Business Media. p. 340.
15047:
Stochastic Flows and Stochastic Differential Equations
14928:
14926:
14882:. Springer Science & Business Media. p. 530.
14760:
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14756:
14649:
14285:. Springer Science & Business Media. p. 104.
14131:. Springer Science & Business Media. p. 528.
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13331:. Springer Science & Business Media. p. 118.
12817:. Springer Science & Business Media. p. 136.
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12492:. Springer Science & Business Media. p. 163.
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12355:. Springer Science & Business Media. p. 408.
12344:
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12285:. Springer Science & Business Media. p. 580.
11848:. Springer Science & Business Media. p. 527.
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11605:
11561:. Springer Science & Business Media. pp. 4â.
10874:
10098:
9796:
Another approach is possible, originally developed by
8906:
the 19th century, with work done chiefly by Clausius,
7050:{\displaystyle \left(Y(t_{1}),\ldots ,Y(t_{n})\right)}
6976:{\displaystyle \left(X(t_{1}),\ldots ,X(t_{n})\right)}
6654:{\displaystyle \{X_{t}\in F{\text{ for all }}t\in G\}}
6513:
1627:, this random walk is called a symmetric random walk.
21925:
Numerical methods for ordinary differential equations
19781:
19734:. Springer Science & Business Media. p. 14.
19670:
19569:. Springer Science & Business Media. p. 11.
19337:
19317:
Marc Barbut; Bernard Locker; Laurent Mazliak (2016).
19190:. Springer Science & Business Media. p. ix.
19179:
19177:
19150:
Charles Miller Grinstead; James Laurie Snell (1997).
19073:. Handbook of Statistics. Vol. 19. p. 367.
18823:
18707:
16927:
16925:
16923:
16517:. Springer Science & Business Media. p. 22.
16463:. Springer Science & Business Media. p. 96.
16450:
16436:. Springer Science & Business Media. p. 11.
16209:
Introduction to Stochastic Calculus with Applications
16153:. Springer Science & Business Media. p. 27.
16126:. Springer Science & Business Media. p. 61.
15872:
15870:
15769:. Springer Science & Business Media. p. 58.
15715:. Springer Science & Business Media. p. 79.
15688:. Springer Science & Business Media. p. 22.
15627:
15488:
Introduction to Stochastic Calculus with Applications
15480:
15478:
15476:
15459:. Springer Science & Business Media. p. 53.
15286:
15220:
15111:. Springer Science & Business Media. p. 11.
15077:. Springer Science & Business Media. p. 35.
14939:. Springer Science & Business Media. p. 14.
14906:
Introduction to Stochastic Calculus with Applications
14875:
14795:
Introduction to Stochastic Calculus with Applications
14563:
14561:
14559:
14557:
14513:. Springer Science & Business Media. p. 10.
14459:. Springer Science & Business Media. p. 23.
14366:. Springer Science & Business Media. p. 10.
14272:
14258:. Springer Science & Business Media. p. 25.
14121:
13945:
13844:. Springer Science & Business Media. p. 19.
13647:
Introduction to Stochastic Calculus with Applications
13547:
Introduction to Stochastic Calculus with Applications
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13304:. Springer Science & Business Media. p. 33.
13264:
13250:. Springer Science & Business Media. p. IX.
13104:
13044:
Introduction to Stochastic Calculus with Applications
12993:. Springer Science & Business Media. p. 88.
12963:
Introduction to Stochastic Calculus with Applications
12955:
12953:
12804:
12616:
Introduction to Stochastic Calculus with Applications
11946:. Springer Science & Business Media. p. 63.
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11078:
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11038:
11036:
11034:
11032:
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10955:. Springer Science & Business Media. p. 42.
10928:. Springer Science & Business Media. p. 21.
10911:
10847:. Springer Science & Business Media. p. 32.
10534:
Baccelli, François; Blaszczyszyn, Bartlomiej (2009).
10465:
10150:. Springer Science & Business Media. p. 29.
9814:
9764:
9744:
9710:
9686:
9657:
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9580:
9556:
9536:
9212:
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8771:
8737:
8681:
8563:
8535:
8483:
8463:
8443:
8306:
8247:
8201:
8084:
8075:
denotes the space of cĂ dlĂ g functions defined on the
8046:
8026:
8020:, so the function space is also referred to as space
8006:
7966:
7934:
7761:
7630:
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2010:
1990:
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1918:
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1607:
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1332:
1305:
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1243:
1223:
1188:
1156:
1110:
1062:
1021:
1001:
864:
840:
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707:
411:
402:
396:
384:
22301:
Société de Mathématiques Appliquées et Industrielles
22294:
Japan Society for Industrial and Applied Mathematics
21930:
Numerical methods for partial differential equations
21644:
19775:
19664:
19583:
19407:
18591:
18483:
17880:
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16673:
16619:
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16423:
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15883:. Springer Science & Business Media. p. 7.
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15742:. Springer Science & Business Media. p. 2.
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15505:
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14696:
14506:
14299:
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14112:
14071:. Springer Science & Business Media. p. 1.
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11894:
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10715:
10713:
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10709:
10707:
10705:
10703:
10654:
10589:
10533:
9467:
derived in a 1928 paper an equation, now called the
9194:
random walks. For example, the problem known as the
8435:
processes with stationary and independent increments
405:
387:
22117:
19643:
19610:
19556:
19504:
19502:
19500:
19413:
19068:
18664:
18462:
18212:
18171:. Cambridge University Press. p. viii and ix.
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17173:
17171:
17071:
16997:Jesper Moller; Rasmus Plenge Waagepetersen (2003).
16646:
16332:
16194:
16167:
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15399:
15397:
15395:
15393:
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14058:
13277:. World Scientific Publishing Co Inc. p. 186.
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12831:
12703:Bertsekas, Dimitri P.; Tsitsiklis, John N. (2002).
12702:
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12633:
12369:
12333:
12000:
11998:
11909:
11832:
11811:
11707:
11420:
11387:
10942:
9960:for the latter in a style similar to, for example,
9399:Markov processes and Markov chains are named after
2588:or a more general set, such as the Cartesian plane
2165:-valued random variables, which can be written as:
1551:A classic example of a random walk is known as the
393:
20211:
19529:
19174:
18628:Random Walks and Random Environments: Random walks
18618:
18218:
18185:
17453:
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11971:
11862:
11195:
11074:
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11068:
10689:
10506:Probability and Statistics for Computer Scientists
10448:Stochastic Optimal Control: The Discrete-Time Case
10373:
10139:
10137:
10135:
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9770:
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8207:
8102:
8067:
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7984:
7952:
7897:
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7612:
7577:
7542:are independent, then they are also uncorrelated.
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2016:
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20206:
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18888:
18886:
18658:Kongelige Danske Videnskabernes Selskabs Skrifter
18380:
18361:
18191:
17640:
16963:
16510:
16504:
16483:
16429:
16420:
16372:
16311:
16302:
15729:
15138:
15136:
15134:
15132:
15130:
15128:
15037:
14425:
14278:
14224:
14085:
13951:
13918:
13777:
13429:
13324:
12923:
12892:
12890:
12888:
12750:
12575:
12422:
12405:
12319:. Cambridge University Press. pp. 121, 122.
11581:
11514:
11222:
11007:
10948:
10918:Gusak, Dmytro; Kukush, Alexander; Kulik, Alexey;
10700:
10687:
10143:
9948:can refer to the physical process, also known as
4799:is a stationary stochastic process, then for any
4742:
4628:
2873:{\displaystyle X(\cdot ,\omega ):T\rightarrow S,}
2004:, all take values in the same mathematical space
569:are used interchangeably, often with no specific
22351:
20668:Stochastic chains with memory of variable length
20137:
19755:Krishna B. Athreya; Soumendra N. Lahiri (2006).
19698:Krishna B. Athreya; Soumendra N. Lahiri (2006).
19536:Krishna B. Athreya; Soumendra N. Lahiri (2006).
19497:
18992:
18949:
18564:
18458:
18456:
18426:
18315:
18280:
17761:
17457:
17329:
17275:
17168:
16969:
16393:
16258:
16205:
16005:
15702:
15484:
15390:
14902:
14791:
14615:
14446:
14091:
14055:
14034:
13864:
13810:
13697:
13643:
13543:
13435:
13040:
12959:
12612:
12454:
12452:
12244:
12032:
11995:
11693:. Cambridge University Press. pp. 121â124.
11575:
11554:
11303:
11255:
10785:
10783:
10781:
10779:
10777:
10775:
10773:
10771:
10598:
10414:Random processes for image and signal processing
10317:
9925:Stochastic processes and boundary value problems
9784:
9435:in 1907, and a branching process, introduced by
9077:based on the Wiener or Brownian motion process.
8522:{\displaystyle 0\leq t_{1}\leq \dots \leq t_{n}}
4415:; it can be viewed as a "projection" of the law
3452:{\displaystyle X\colon \Omega \rightarrow S^{T}}
1984:; and the random variables, indexed by some set
1641:The Wiener process is a stochastic process with
1448:One of the simplest stochastic processes is the
902:word meaning "to aim at a mark, guess", and the
20039:
19802:
19508:
19476:
18794:
18753:
18624:
18537:
18520:Elements of Random Walk and Diffusion Processes
18248:"Kolmogorov and the Theory of Markov Processes"
18072:
18070:
17448:
17415:
17413:
17411:
17409:
17407:
17405:
17366:Chung, Kai Lai (1998). "Probability and Doob".
16791:
16737:
16456:
16399:
15984:
15789:
15735:
15681:
15573:
15567:
15319:
15199:
15070:
15016:
14771:. Cambridge University Press. pp. 93, 94.
14676:
14628:
14479:
13984:. Cambridge University Press. pp. 10, 18.
13718:
13462:
12810:
12733:Elements of Random Walk and Diffusion Processes
12554:
12378:
12069:Alexandr A. Chuprov: Life, Work, Correspondence
11414:
11282:
11228:
11141:
11065:
10986:
10132:
5281:{\displaystyle \{{\mathcal {F}}_{t}\}_{t\in T}}
5100:and the covariance of the two random variables
4341:is the joint distribution of the random vector
2698:. This mathematical space can be defined using
2637:-dimensional Euclidean space, where an element
22289:Society for Industrial and Applied Mathematics
20182:
20165:Introduction to the Theory of Random Processes
20102:
20081:
19808:
19650:. Wiley India Pvt. Limited. pp. 493â494.
19589:
19562:
19364:
19284:Bru, B.; Hertz, S. (2001). "Maurice Fréchet".
19183:
18883:
18597:
18489:
18195:Stochastic Calculus and Financial Applications
18008:
18006:
18004:
17558:
17556:
17554:
17552:
17550:
17548:
17546:
17544:
17542:
17540:
17308:
17234:
17147:
17050:
16872:
16514:Stochastic Calculus and Financial Applications
16487:Stochastic Calculus and Financial Applications
16433:Stochastic Calculus and Financial Applications
16376:Stochastic Calculus and Financial Applications
16315:Stochastic Calculus and Financial Applications
16264:
16173:
15903:
15876:
15816:
15514:
15325:
15263:. Wiley India Pvt. Limited. pp. 526â527.
15226:
15173:Introduction to the Theory of Random Processes
15125:
15043:
14962:
14932:
14764:
14736:Introduction to the Theory of Random Processes
14705:
14685:
14598:Introduction to the Theory of Random Processes
14359:
14305:
13977:
13328:Stochastic Calculus and Financial Applications
13156:
13133:Introduction to the Theory of Random Processes
13061:
12885:
12837:
12642:
12512:
12428:
12348:
12152:
11873:
11724:
11474:
11447:
11174:
10681:
10664:Introduction to the Theory of Random Processes
10568:Stochastic Calculus and Financial Applications
10379:
10233:Introduction to the Theory of Random Processes
10179:
10147:Stochastic Calculus and Financial Applications
10087:
8135:
6661:differ from each other at most on a subset of
4408:{\displaystyle (X({t_{1}}),\dots ,X({t_{n}}))}
2294:. A stochastic process can also be written as
2071:In other words, for a given probability space
443:of the family often has the interpretation of
21540:
20257:
19984:
19277:
19209:
19207:
18516:
18510:
18453:
18139:
18122:
18120:
18118:
18116:
18114:
17960:
17921:
17919:
17361:
17359:
16564:
16281:
16066:Reuven Y. Rubinstein; Dirk P. Kroese (2011).
15957:
15383:
15381:
15379:
15149:. American Mathematical Soc. pp. 32â33.
14567:
14064:
13729:
13537:
13361:. Cambridge University Press. pp. 1, 3.
12980:
12723:
12585:
12548:
12449:
12038:
11548:
11495:
11468:
11044:
10768:
10723:
10435:
10404:
10383:Stochastic Processes: From Physics to Finance
10311:
10290:Stochastic Processes in Physics and Chemistry
10186:. Courier Dover Publications. pp. 7, 8.
8995:Grundbegriffe der Wahrscheinlichkeitsrechnung
5668:, resulting in finer and finer partitions of
4735:
4722:
4654:
4635:
926:first appeared in English in a 1934 paper by
585:. If the random variables are indexed by the
335:
22107:Supersymmetric theory of stochastic dynamics
20061:Robert J. Adler; Jonathan E. Taylor (2009).
19728:Robert J. Adler; Jonathan E. Taylor (2009).
19616:
18164:
18067:
17402:
16693:. Cambridge University Press. p. viii.
16686:
16147:Donald L. Snyder; Michael I. Miller (2012).
16119:
15823:. John Wiley & Sons. pp. 373, 374.
15762:
15601:Nicholas H. Bingham; RĂŒdiger Kiesel (2013).
14712:. John Wiley & Sons. pp. 298, 299.
14252:Donald L. Snyder; Michael I. Miller (2012).
13298:Donald L. Snyder; Michael I. Miller (2012).
12864:
12756:
12238:
11782:Donald L. Snyder; Michael I. Miller (2012).
11731:. John Wiley & Sons. pp. 294, 295.
11168:
11121:Robert J. Adler; Jonathan E. Taylor (2009).
10841:Donald L. Snyder; Michael I. Miller (2012).
10342:
10250:
10175:
10173:
10171:
10169:
10167:
9499:
9155:Discoveries of specific stochastic processes
7996:. A Skorokhod function space, introduced by
6648:
6615:
6595:
6556:
5637:is known or available, which is captured in
5263:
5245:
4539:{\displaystyle S^{n}=S\times \dots \times S}
4043:Finite-dimensional probability distributions
4006:
3973:
3905:
3872:
3088:
3061:
2950:
2917:
2799:is a stochastic process, then for any point
2786:
2753:
2334:
2301:
2201:
2174:
1204:
1189:
1170:
1157:
1125:
1111:
1079:
1063:
20116:
19623:. Cambridge University Press. p. 410.
19596:. Cambridge University Press. p. 124.
19444:Bulletin of the London Mathematical Society
19234:
19103:
18645:
18570:
18543:
18432:
18321:
18239:
18222:Martingale Limit Theory and Its Application
18001:
17821:Bulletin of the London Mathematical Society
17537:
17281:
16656:Martingale Limit Theory and Its Application
16538:Geoffrey Grimmett; David Stirzaker (2001).
16349:Martingale Limit Theory and Its Application
16275:
15910:. Courier Dover Publications. p. 188.
14855:. Cambridge University Press. p. 130.
14540:. Cambridge University Press. p. 123.
14405:. Cambridge University Press. p. 571.
14014:. John Wiley & Sons. pp. 41, 108.
13523:. Cambridge University Press. p. 131.
13274:A First Look at Rigorous Probability Theory
13014:Geoffrey Grimmett; David Stirzaker (2001).
12936:. Cambridge University Press. p. 454.
12929:
12871:. Cambridge University Press. p. 277.
12435:. Cambridge University Press. p. 383.
12279:Bert E. Fristedt; Lawrence F. Gray (2013).
12201:
12095:
12093:
12091:
12089:
11913:A First Look at Rigorous Probability Theory
11441:
11336:
11085:. Cambridge University Press. p. 175.
9423:In 1912, Poincaré studied Markov chains on
8828:
8789:is not preferred, as historically the word
518:. Furthermore, seemingly random changes in
21547:
21533:
20796:Autoregressiveâmoving-average (ARMA) model
20264:
20250:
19483:. Cambridge University Press. p. 67.
19204:
18111:
17916:
17889:
17853:
17726:
17356:
17335:
17213:
17098:
16879:. Cambridge University Press. p. 10.
16798:. Cambridge University Press. p. 19.
16768:Theory of Probability and Random Processes
16744:. Cambridge University Press. p. 69.
16626:Francois Baccelli; Pierre Bremaud (2013).
16018:. Cambridge University Press. p. 19.
15708:
15580:. Cambridge University Press. p. 34.
15403:
15376:
15142:
15108:Mathematical Methods for Financial Markets
15050:. Cambridge University Press. p. 31.
15023:. Cambridge University Press. p. 20.
14993:Continuous Martingales and Brownian Motion
14825:. Cambridge University Press. p. 37.
14798:. Imperial College Press. pp. 22â23.
14510:Continuous Martingales and Brownian Motion
14452:
13247:Continuous Martingales and Brownian Motion
12760:Probability Theory: A Comprehensive Course
12458:
12158:
10623:
10558:
10286:
9680:a sample function of a stochastic process
9627:a sample function of a stochastic process
9574:a sample function of a stochastic process
7811:
7807:
7399:
7395:
6798:{\displaystyle (\Omega ,{\mathcal {F}},P)}
6425:{\displaystyle \Omega _{0}\subset \Omega }
6284:other conditions, its index set must be a
6061:{\displaystyle (\Omega ,{\mathcal {F}},P)}
5323:{\displaystyle (\Omega ,{\mathcal {F}},P)}
4993:{\displaystyle X_{t_{1}},\dots X_{t_{n}},}
3494:{\displaystyle (\Omega ,{\mathcal {F}},P)}
2145:, a stochastic process is a collection of
2106:{\displaystyle (\Omega ,{\mathcal {F}},P)}
1881:{\displaystyle (\Omega ,{\mathcal {F}},P)}
342:
328:
20088:. Springer Science & Business Media.
20067:. Springer Science & Business Media.
19930:
19899:
19881:Learn how and when to remove this message
19542:. Springer Science & Business Media.
19370:
18898:
18814:
18800:
18778:
18759:
18716:
18263:
18094:
18026:
17984:
17943:
17689:
17603:
17580:
17494:
17258:
17240:
17084:. Springer Science & Business Media.
16876:Stochastic Geometry for Wireless Networks
16718:
16632:. Springer Science & Business Media.
16578:
16093:Dani Gamerman; Hedibert F. Lopes (2006).
15363:
14486:. Cambridge University Press. p. 4.
14339:. Wiley India Pvt. Limited. p. 493.
14196:
13981:Stochastic Geometry for Wireless Networks
13597:
13110:
12986:
12897:Weiss, George H. (2006). "Random Walks".
12790:. Cambridge University Press. p. 1.
12561:. Cambridge University Press. p. 1.
12516:Fourier Analysis and Stochastic Processes
12479:
12465:. American Mathematical Soc. p. 13.
12135:
12125:
12066:Oscar Sheynin; Heinrich Strecker (2011).
11940:Peter E. Kloeden; Eckhard Platen (2013).
11877:Fourier Analysis and Stochastic Processes
11818:. Wiley India Pvt. Limited. p. 482.
11664:
11501:
11451:Fourier Analysis and Stochastic Processes
11400:. Springer Science & Business Media.
11330:
11276:
11262:. Springer Science & Business Media.
10789:
10733:
10602:Martingale Methods in Financial Modelling
10599:Musiela, Marek; Rutkowski, Marek (2006).
10537:Stochastic Geometry and Wireless Networks
10496:
10441:
10410:
10380:Paul, Wolfgang; Baschnagel, Jörg (2013).
10348:
10280:
10256:
10164:
10118:. Cambridge University Press. p. 1.
10005:Also known as James or Jacques Bernoulli.
9738:can be read "a member of the set", as in
9065:published papers developing the field of
8914:, which would later have an influence on
6843:
5466:denotes the total order of the index set
3101:is a stochastic process with state space
2597:
1024:
910:, originally published in Latin in 1713,
20271:
19283:
19071:Stochastic Processes: Theory and Methods
17966:
17243:"Decoding Cardano's Liber de Ludo Aleae"
17024:Samuel Karlin; Howard E. Taylor (2012).
16931:
16849:Stochastic Geometry and Its Applications
16822:Stochastic Geometry and Its Applications
16238:Ioannis Karatzas; Steven Shreve (1991).
16150:Random Point Processes in Time and Space
16039:Samuel Karlin; Howard E. Taylor (2012).
15931:Samuel Karlin; Howard E. Taylor (2012).
15847:Samuel Karlin; Howard E. Taylor (2012).
14849:L. C. G. Rogers; David Williams (2000).
14659:Stochastic Geometry and Its Applications
14534:L. C. G. Rogers; David Williams (2000).
14255:Random Point Processes in Time and Space
14011:Stochastic Geometry and Its Applications
13892:Samuel Karlin; Howard E. Taylor (2012).
13671:Ioannis Karatzas; Steven Shreve (1991).
13617:Samuel Karlin; Howard E. Taylor (2012).
13571:Ioannis Karatzas; Steven Shreve (1991).
13490:Ioannis Karatzas; Steven Shreve (1991).
13409:Ioannis Karatzas; Steven Shreve (1991).
13382:Ioannis Karatzas; Steven Shreve (1991).
13301:Random Point Processes in Time and Space
13214:Ioannis Karatzas; Steven Shreve (1991).
13187:Samuel Karlin; Howard E. Taylor (2012).
12672:
12485:
12313:L. C. G. Rogers; David Williams (2000).
12086:
11785:Random Point Processes in Time and Space
11755:Samuel Karlin; Howard E. Taylor (2012).
11687:L. C. G. Rogers; David Williams (2000).
11619:Samuel Karlin; Howard E. Taylor (2012).
11310:Samuel Karlin; Howard E. Taylor (2012).
11202:L. C. G. Rogers; David Williams (2000).
10844:Random Point Processes in Time and Space
10112:L. C. G. Rogers; David Williams (2000).
9978:sense), in addition to other conditions.
9258:
9038:
8958:. In 1925, another French mathematician
8932:International Congress of Mathematicians
8888:
8293:{\displaystyle X_{1},X_{2},X_{3},\dots }
6902:{\displaystyle t_{1},\ldots ,t_{n}\in T}
6228:are modifications of each other and are
5330:is a family of sigma-algebras such that
3159:, then for any two non-negative numbers
1663:
739:
353:
19371:Bernstein, Jeremy (2005). "Bachelier".
18386:
18076:
17419:
17312:Probability Theory: A Historical Sketch
17220:. Infobase Publishing. pp. 24â26.
17151:Probability Theory: A Historical Sketch
16765:Leonid Koralov; Yakov G. Sinai (2007).
16241:Brownian Motion and Stochastic Calculus
14172:
13674:Brownian Motion and Stochastic Calculus
13650:. Imperial College Press. p. 124.
13574:Brownian Motion and Stochastic Calculus
13493:Brownian Motion and Stochastic Calculus
13412:Brownian Motion and Stochastic Calculus
13385:Brownian Motion and Stochastic Calculus
13217:Brownian Motion and Stochastic Calculus
12784:Gregory F. Lawler; Vlada Limic (2010).
12385:. Oxford University Press. p. 45.
12282:A Modern Approach to Probability Theory
11148:Gregory F. Lawler; Vlada Limic (2010).
10472:Thomas M. Cover; Joy A. Thomas (2012).
10349:Laing, Carlo; Lord, Gabriel J. (2010).
9104:. Earlier work had been carried out by
9056:Stochastic processes after World War II
8437:. In other words, a stochastic process
8239:independent and identically distributed
4142:{\displaystyle t_{1},\dots ,t_{n}\in T}
2956:{\displaystyle \{X(t,\omega ):t\in T\}}
2792:{\displaystyle \{X(t,\omega ):t\in T\}}
2340:{\displaystyle \{X(t,\omega ):t\in T\}}
1575:, or decreases by one with probability
1454:independent and identically distributed
681:. Historically, the index set was some
14:
22352:
21102:Doob's martingale convergence theorems
19912:
19480:LĂ©vy Processes and Stochastic Calculus
19240:
19156:. American Mathematical Soc. pp.
19109:
18651:
18604:. John Wiley & Sons. p. 385.
18577:. John Wiley & Sons. p. 202.
18496:. John Wiley & Sons. p. 374.
18439:. John Wiley & Sons. p. 226.
18245:
18012:
17562:
17288:. John Wiley & Sons. p. 221.
17177:
17027:A First Course in Stochastic Processes
16852:. John Wiley & Sons. p. 108.
16825:. John Wiley & Sons. p. 109.
16795:LĂ©vy Processes and Stochastic Calculus
16741:LĂ©vy Processes and Stochastic Calculus
16212:. Imperial College Press. p. 65.
16072:. John Wiley & Sons. p. 225.
16042:A First Course in Stochastic Processes
16015:Markov Chains and Stochastic Stability
16012:Sean Meyn; Richard L. Tweedie (2009).
15934:A First Course in Stochastic Processes
15850:A First Course in Stochastic Processes
15739:Basics of Applied Stochastic Processes
15553:. John Wiley & Sons. p. 121.
15020:LĂ©vy Processes and Stochastic Calculus
14969:. John Wiley & Sons. p. 472.
14909:. Imperial College Press. p. 48.
14662:. John Wiley & Sons. p. 112.
14483:LĂ©vy Processes and Stochastic Calculus
14312:. John Wiley & Sons. p. 296.
14181:"Borel structures for function spaces"
14178:
13895:A First Course in Stochastic Processes
13787:An Introduction to Stochastic Modeling
13784:Mark A. Pinsky; Samuel Karlin (2011).
13620:A First Course in Stochastic Processes
13190:A First Course in Stochastic Processes
13163:. John Wiley & Sons. p. 471.
13047:. Imperial College Press. p. 56.
12966:. Imperial College Press. p. 81.
12844:. John Wiley & Sons. p. 383.
12679:. John Wiley & Sons. p. 301.
12649:. John Wiley & Sons. p. 293.
12619:. Imperial College Press. p. 55.
12072:. V&R unipress GmbH. p. 136.
11758:A First Course in Stochastic Processes
11622:A First Course in Stochastic Processes
11313:A First Course in Stochastic Processes
11232:LĂ©vy Processes and Stochastic Calculus
10629:
10564:
9520:
9351:, due to its definition involving the
9163:
6763:defined on the same probability space
6346:{\displaystyle (\Omega ,{\cal {F}},P)}
6026:defined on the same probability space
5795:{\displaystyle (\Omega ,{\cal {F}},P)}
3410:
2694:of a stochastic process is called its
922:with a sense meaning random. The term
765:of the index set and the state space.
358:A computer-simulated realization of a
21528:
20854:Constant elasticity of variance (CEV)
20844:ChanâKarolyiâLongstaffâSanders (CKLS)
20245:
20019:
19758:Measure Theory and Probability Theory
19701:Measure Theory and Probability Theory
19539:Measure Theory and Probability Theory
19213:
19001:Archive for History of Exact Sciences
18998:
18958:Archive for History of Exact Sciences
18955:
18550:. John Wiley & Sons. p. 63.
18341:Malte Henkel; Dragi Karevski (2012).
18286:
18126:
17925:
17770:Archive for History of Exact Sciences
17767:
17732:
17692:Archive for History of Exact Sciences
17649:Archive for History of Exact Sciences
17646:
17606:Archive for History of Exact Sciences
17365:
17104:
16069:Simulation and the Monte Carlo Method
15964:. Springer-Verlag. pp. 106â121.
15491:. Imperial College Press. p. 4.
15371:A Foundation in Digital Communication
14574:. Springer-Verlag. pp. 6 and 7.
13898:. Academic Press. pp. 118, 119.
13070:Archive for History of Exact Sciences
13067:
12896:
12736:. John Wiley & Sons. p. 11.
12102:"Stochastic Processes and Statistics"
11394:D.J. Daley; David Vere-Jones (2007).
10502:
8926:Measure theory and probability theory
7498:Independence implies uncorrelatedness
5610:passes, more and more information on
4334:{\displaystyle \mu _{t_{1},..,t_{n}}}
3598:is a probability measure, the symbol
1838:
1701:-dimensional Euclidean space. If the
918:who in 1917 wrote in German the word
19837:
19815:. John Wiley & Sons. p. 5.
18670:
18601:Probability and Stochastic Processes
18523:. John Wiley & Sons. p. 5.
18493:Probability and Stochastic Processes
18469:. North-Holland Pub. pp. 8â10.
17892:Encyclopedia of Statistical Sciences
17856:Encyclopedia of Statistical Sciences
17500:
17183:Encyclopedia of Statistical Sciences
15820:Probability and Stochastic Processes
15176:. Courier Corporation. p. 150.
14966:Probability and Stochastic Processes
14709:Probability and Stochastic Processes
14309:Probability and Stochastic Processes
13704:. John Wiley & Sons. p. 2.
13160:Probability and Stochastic Processes
12899:Encyclopedia of Statistical Sciences
12841:Probability and Stochastic Processes
12676:Probability and Stochastic Processes
12646:Probability and Stochastic Processes
12266:participating institution membership
12099:
12026:participating institution membership
11728:Probability and Stochastic Processes
11588:Stochastic Processes in Cell Biology
10263:Stochastic Processes in Cell Biology
9342:
7613:{\displaystyle \left\{Y_{t}\right\}}
7578:{\displaystyle \left\{X_{t}\right\}}
7128:{\displaystyle \left\{Y_{t}\right\}}
7093:{\displaystyle \left\{X_{t}\right\}}
5981:
5559:. The intuition behind a filtration
3242:{\displaystyle t_{2}\in [0,\infty )}
3197:{\displaystyle t_{1}\in [0,\infty )}
1437:
788:continuous-time stochastic processes
533:or Brownian motion process, used by
18225:. Elsevier Science. pp. 1, 2.
18146:"Kiyosi ItĂŽ receives Kyoto Prize".
17342:. Infobase Publishing. p. 37.
17181:(2006). "Probability, History of".
15937:. Academic Press. pp. 29, 30.
15550:Convergence of Probability Measures
15373:, Cambridge University Press, 2009.
14819:Peter Mörters; Yuval Peres (2010).
13960:. Oxford University Press. p.
13733:A First Course in Stochastic Models
13517:Peter Mörters; Yuval Peres (2010).
13355:Peter Mörters; Yuval Peres (2010).
13193:. Academic Press. pp. 21, 22.
13136:. Courier Corporation. p. 21.
12729:
12414:. Oxford University Press. p.
9839:List of stochastic processes topics
9394:
9347:The Poisson process is named after
9238:
8130:
8000:, is often denoted with the letter
7060:
5539:, which can be interpreted as time
5074:has a finite second moment for all
3569:{\displaystyle \mu =P\circ X^{-1},}
3336:{\displaystyle X_{t_{2}}-X_{t_{1}}}
1384:, then one can write, for example,
812:process with continuous state space
24:
21556:Industrial and applied mathematics
21341:Skorokhod's representation theorem
21122:Law of large numbers (weak/strong)
19833:
19042:D.J. Daley; D. Vere-Jones (2006).
18406:D.J. Daley; D. Vere-Jones (2006).
17078:D.J. Daley; D. Vere-Jones (2006).
16900:D.J. Daley; D. Vere-Jones (2006).
15146:Essentials of Stochastic Processes
14739:. Courier Corporation. p. 8.
14601:. Courier Corporation. p. 4.
14429:Adventures in Stochastic Processes
13838:D.J. Daley; D. Vere-Jones (2006).
13757:D.J. Daley; D. Vere-Jones (2006).
12787:Random Walk: A Modern Introduction
12709:. Athena Scientific. p. 273.
12462:Essentials of Stochastic Processes
11367:G. Latouche; V. Ramaswami (1999).
11151:Random Walk: A Modern Introduction
10352:Stochastic Methods in Neuroscience
10236:. Courier Corporation. p. 1.
9481:infinitely divisible distributions
8989:Birth of modern probability theory
8697:
7994:continue Ă droite, limite Ă gauche
7976:
7909:
7869:
7814:
7683:
7632:
7457:
7402:
7198:
7147:
6781:
6773:
6669:
6532:
6526:
6446:
6419:
6407:
6329:
6321:
6044:
6036:
5941:Instead of modification, the term
5778:
5770:
5675:
5661:{\displaystyle {\mathcal {F}}_{t}}
5647:
5583:{\displaystyle {\mathcal {F}}_{t}}
5569:
5373:
5357:
5340:
5306:
5298:
5251:
5214:, which has other names including
5210:introduced the related concept of
4911:{\displaystyle t_{1},\dots ,t_{n}}
4646:
3982:
3881:
3477:
3469:
3433:
3233:
3188:
3143:
2982:{\displaystyle \omega \in \Omega }
2976:
2818:{\displaystyle \omega \in \Omega }
2812:
2733:
2392:{\displaystyle \omega \in \Omega }
2386:
2129:
2089:
2081:
2055:
1921:
1895:
1864:
1856:
1767:
1429:to denote the stochastic process.
1368:
1143:{\displaystyle \{X_{t}\}_{t\in T}}
756:
25:
22381:
21786:Stochastic differential equations
21311:Martingale representation theorem
20223:
17503:The American Mathematical Monthly
17368:The American Mathematical Monthly
16460:Foundations of Modern Probability
15074:Foundations of Modern Probability
13466:Foundations of Modern Probability
12814:Foundations of Modern Probability
11534:. European Mathematical Society.
11051:. Springer-Verlag. pp. 1â2.
10990:Foundations of Modern Probability
9474:
9096:In 1953, Doob published his book
7752:is zero for all times. Formally:
7340:is zero for all times. Formally:
6850:{\displaystyle n\in \mathbb {N} }
5921:{\displaystyle P(X_{t}=Y_{t})=1,}
3722:is the space of all the possible
3618:denotes function composition and
3459:defined on the probability space
2456:is the space of all the possible
1630:
1097:{\displaystyle \{X(t)\}_{t\in T}}
804:integer-valued stochastic process
721:-dimensional Euclidean space. An
22102:Supersymmetric quantum mechanics
21356:Stochastic differential equation
21246:Doob's optional stopping theorem
21241:DoobâMeyer decomposition theorem
20229:
19962:10.1111/j.1751-5823.2012.00181.x
19950:International Statistical Review
19842:
19620:Probability: Theory and Examples
18945:from the original on 2018-07-21.
18879:from the original on 2018-07-21.
18790:from the original on 2011-06-05.
18631:. Clarendon Press. p. 111.
17472:10.1111/j.1751-5823.2007.00009.x
17460:International Statistical Review
17315:. Elsevier Science. p. 56.
16541:Probability and Random Processes
16140:
14245:
14179:Aumann, Robert (December 1961).
13291:
13017:Probability and Random Processes
12868:Probability: Theory and Examples
11079:LoĂŻc Chaumont; Marc Yor (2012).
10889:10.1111/j.1751-5823.2012.00181.x
10877:International Statistical Review
10044:
10035:
10026:
10017:
10008:
9999:
9990:
9981:
9971:
8754:
7824:
7819:
7642:
7637:
7545:
7412:
7407:
7157:
7152:
6467:{\displaystyle P(\Omega _{0})=0}
5822:is said to be a modification of
2610:{\displaystyle \mathbb {R} ^{2}}
2210:{\displaystyle \{X(t):t\in T\}.}
1237:. Some authors mistakenly write
1037:{\displaystyle \mathbb {R} ^{n}}
380:
48:
21984:Stochastic variational calculus
21776:Ordinary differential equations
21226:Convergence of random variables
21112:FisherâTippettâGnedenko theorem
17928:"Obituary: Joseph Leonard Doob"
17900:10.1002/0471667196.ess6027.pub2
17864:10.1002/0471667196.ess6027.pub2
17191:10.1002/0471667196.ess2065.pub2
17154:. Elsevier Science. p. 1.
16659:. Elsevier Science. p. x.
16352:. Elsevier Science. p. 2.
14990:Daniel Revuz; Marc Yor (2013).
14507:Daniel Revuz; Marc Yor (2013).
14185:Illinois Journal of Mathematics
14041:. Clarendon Press. p. 11.
13871:. Clarendon Press. p. 22.
13817:. Clarendon Press. p. 38.
13790:. Academic Press. p. 241.
13623:. Academic Press. p. 340.
12907:10.1002/0471667196.ess2180.pub2
12382:Stochastic Processes and Models
11775:
11531:Surveys in Stochastic Processes
11249:
10834:
10638:Springer Science+Business Media
10607:Springer Science+Business Media
10573:Springer Science+Business Media
10388:Springer Science+Business Media
9938:
9854:Dynamics of Markovian particles
9404:vector of values, so proving a
9180:
8720:
8421:
8412:martingale convergence theorems
8225:Martingale (probability theory)
7868:
7812:
7806:
7456:
7400:
7394:
6857:and for every choice of epochs
6718:
6474:, such that for every open set
6275:
5691:
5220:stationarity in the broad sense
4765:
4095:finite-dimensional distribution
4049:Finite-dimensional distribution
3282:{\displaystyle t_{1}\leq t_{2}}
3094:{\displaystyle \{X(t):t\in T\}}
2882:is called a sample function, a
2544:or an interval, giving the set
2429:-valued random variable, where
1422:{\displaystyle (X_{t},t\geq 0)}
671:
21781:Partial differential equations
21654:Arbitrary-precision arithmetic
20824:Binomial options pricing model
20192:. Courier Dover Publications.
19987:A Festschrift for Herman Rubin
19782:Alexander A. Borovkov (2013).
19671:Alexander A. Borovkov (2013).
19323:. Springer London. p. 5.
19286:Statisticians of the Centuries
19124:10.1080/03461238.1969.10404602
19112:Scandinavian Actuarial Journal
18368:"2006 Fields Medals Awarded".
18301:10.1080/03461238.1995.10413952
18289:Scandinavian Actuarial Journal
18083:Journal of Applied Probability
17932:Journal of Applied Probability
17030:. Academic Press. p. 31.
16976:. Clarendon Press. p. 8.
16288:. Cambridge University Press.
16045:. Academic Press. p. 47.
15880:Applied Probability and Queues
15853:. Academic Press. p. 49.
15628:Alexander A. Borovkov (2013).
15518:Applied Probability and Queues
15287:Alexander A. Borovkov (2013).
14876:Alexander A. Borovkov (2013).
14098:. Clarendon Press. p. v.
12990:Probability: A Graduate Course
12592:. Springer-Verlag. p. 3.
12352:Applied Probability and Queues
11842:Alexander A. Borovkov (2013).
11761:. Academic Press. p. 26.
11625:. Academic Press. p. 27.
11524:Jochen Blath; Peter Imkeller;
11481:. Cambridge University Press.
11343:. Cambridge University Press.
11340:Random Processes for Engineers
11259:Lectures on Gaussian Processes
11235:. Cambridge University Press.
11208:. Cambridge University Press.
11181:. Cambridge University Press.
11154:. Cambridge University Press.
10726:A Festschrift for Herman Rubin
10475:Elements of Information Theory
9828:-dimensional Euclidean space.
9510:Kolmogorov's existence theorem
9366:Another discovery occurred in
8700:
8688:
8097:
8085:
8062:
8050:
7979:
7967:
7947:
7935:
7859:
7833:
7808:
7739:
7730:
7717:
7708:
7695:
7689:
7677:
7651:
7447:
7421:
7396:
7317:
7304:
7288:
7275:
7259:
7246:
7230:
7217:
7192:
7166:
7039:
7026:
7011:
6998:
6965:
6952:
6937:
6924:
6792:
6770:
6708:-dimensional Euclidean space.
6535:
6520:
6455:
6442:
6353:is separable if its index set
6340:
6318:
6164:
6124:
6055:
6033:
5906:
5880:
5802:as another stochastic process
5789:
5767:
5317:
5295:
5212:stationarity in the wide sense
4717:
4711:
4682:
4676:
4617:
4611:
4402:
4399:
4384:
4369:
4354:
4348:
4258:
4254:
4239:
4224:
4209:
4203:
4009:
3997:
3991:
3970:
3961:
3955:
3896:
3890:
3866:
3860:
3488:
3466:
3436:
3236:
3224:
3191:
3179:
3146:
3134:
3073:
3067:
2963:. This means that for a fixed
2935:
2923:
2861:
2852:
2840:
2771:
2759:
2666:
2319:
2307:
2261:
2255:
2186:
2180:
2132:
2120:
2100:
2078:
1929:{\displaystyle {\mathcal {F}}}
1875:
1853:
1833:
1529:
1416:
1391:
1371:
1359:
1286:
1280:
1253:
1247:
1201:
1195:
1075:
1069:
961:
808:real-valued stochastic process
537:to study price changes on the
115:Collectively exhaustive events
27:Collection of random variables
13:
1:
21669:Interactive geometry software
21291:Kolmogorov continuity theorem
21127:Law of the iterated logarithm
20043:The Geometry of Random Fields
19865:and help improve the section.
19512:The Geometry of Random Fields
19079:10.1016/S0169-7161(01)19014-0
18219:P. Hall; C. C. Heyde (2014).
16653:P. Hall; C. C. Heyde (2014).
15453:Vladimir I. Bogachev (2007).
14632:The Geometry of Random Fields
14399:; Nicolas B. Victoir (2010).
11591:. Springer. pp. viiâix.
11286:The Geometry of Random Fields
10411:Dougherty, Edward R. (1999).
10057:
9885:Stochastic cellular automaton
9785:Resolving construction issues
8706:{\displaystyle I=[0,\infty )}
8218:
8117:
6373:has a dense countable subset
5951:Kolmogorov continuity theorem
5225:
3152:{\displaystyle T=[0,\infty )}
1377:{\displaystyle T=[0,\infty )}
1015:-dimensional Euclidean space
451:processes are widely used as
21296:Kolmogorov extension theorem
20975:Generalized queueing network
20483:Interacting particle systems
19644:Patrick Billingsley (2008).
19593:Probability with Martingales
19414:William J. Anderson (2012).
19294:10.1007/978-1-4613-0179-0_71
18920:10.1080/09672567.2010.540343
18652:Thiele, Thorwald N. (1880).
18463:Joel Louis Lebowitz (1984).
16285:Martingales in Banach Spaces
15991:. Wiley. pp. 174, 231.
15796:. Wiley. pp. 235, 358.
15547:Patrick Billingsley (2013).
15257:Patrick Billingsley (2008).
14768:Probability with Martingales
14333:Patrick Billingsley (2008).
13271:Jeffrey S Rosenthal (2006).
11910:Jeffrey S Rosenthal (2006).
11812:Patrick Billingsley (2008).
11427:. Wiley India Pvt. Limited.
11421:Patrick Billingsley (2008).
11178:Probability with Martingales
10922:; Pilipenko, Andrey (2010).
9069:, which involves stochastic
8127:functions being continuous.
7734:
7502:If two stochastic processes
5720:that has the same index set
3044:
2503:
2245:had the meaning of time, so
885:
744:A single computer-simulated
7:
21721:Computational number theory
21684:Numerical-analysis software
20428:Continuous-time random walk
19893:
19373:American Journal of Physics
19153:Introduction to Probability
18803:"The Theory of Speculation"
18762:"Théorie de la spéculation"
16544:. OUP Oxford. p. 336.
16271:. Wiley. pp. 292, 293.
15655:Davar Khoshnevisan (2006).
15343:Davar Khoshnevisan (2006).
12706:Introduction to Probability
10565:Steele, J. Michael (2001).
10421:Optical Engineering Press.
9875:Interacting particle system
9831:
9469:ChapmanâKolmogorov equation
8404:compensated Poisson process
8170:discrete or continuous time
8148:discrete or continuous time
8136:Markov processes and chains
7985:{\displaystyle [0,\infty )}
7624:if their cross-correlation
6681:{\displaystyle \Omega _{0}}
5160:depends only on the number
2138:{\displaystyle (S,\Sigma )}
1432:
1051:
589:or some higher-dimensional
10:
22386:
21436:Extreme value theory (EVT)
21236:Doob decomposition theorem
20528:OrnsteinâUhlenbeck process
20299:Chinese restaurant process
20218:. Oxford University Press.
20064:Random Fields and Geometry
19731:Random Fields and Geometry
19344:Valeriy Skorokhod (2005).
18899:Jovanovic, Franck (2012).
18770:Ann. Sci. Ăc. Norm. SupĂ©r.
18192:J. Michael Steele (2012).
16597:10.1214/088342306000000088
16511:J. Michael Steele (2012).
16484:J. Michael Steele (2012).
16430:J. Michael Steele (2012).
16373:J. Michael Steele (2012).
16312:J. Michael Steele (2012).
14426:Sidney I. Resnick (2013).
14279:Valeriy Skorokhod (2005).
14225:Valeriy Skorokhod (2005).
13952:Murray Rosenblatt (1962).
13919:Leonard Kleinrock (1976).
13550:. Imperial College Press.
13325:J. Michael Steele (2012).
13220:. Springer. p. VIII.
13020:. OUP Oxford. p. 71.
12406:Murray Rosenblatt (1962).
11124:Random Fields and Geometry
10949:Valeriy Skorokhod (2005).
10696:. Oxford University Press.
10688:Murray Rosenblatt (1962).
10630:Shreve, Steven E. (2004).
10287:Van Kampen, N. G. (2011).
10144:J. Michael Steele (2012).
9861:(for a stochastic process)
8918:'s mathematical model for
8823:
8758:
8724:
8425:
8222:
8139:
7913:
7139:if their cross-covariance
6493:{\displaystyle G\subset T}
6392:{\displaystyle U\subset T}
4769:
4458:For any measurable subset
4046:
2910:of the stochastic process
2906:is interpreted as time, a
2584:can be another set with a
1771:
1634:
1533:
1476:and zero with probability
1441:
1268:abuse of function notation
856:dimensional vector process
18:Law (stochastic processes)
22319:
22281:
22265:
22227:
22179:
22127:Algebra of physical space
21992:
21950:
21744:
21706:
21594:Automated theorem proving
21562:
21504:
21408:
21316:Optional stopping theorem
21213:
21175:
21117:Large deviation principle
21084:
20998:
20955:
20922:
20869:HeathâJarrowâMorton (HJM)
20814:
20806:Moving-average (MA) model
20791:Autoregressive (AR) model
20771:
20681:
20616:Hidden Markov model (HMM)
20598:
20550:SchrammâLoewner evolution
20354:
20279:
20123:. John Wiley & Sons.
19919:The Annals of Probability
18807:Ann. Sci. Ăc. Norm. SupĂ©r
18660:. Series 5 (12): 381â408.
18252:The Annals of Probability
18079:"Doob: a half-century on"
18015:The Annals of Probability
17973:The Annals of Probability
17967:Lindvall, Torgny (1991).
17926:Snell, J. Laurie (2005).
17747:10.1080/00033795800200147
17569:The Annals of Probability
17241:Bellhouse, David (2005).
16970:J. F. C. Kingman (1992).
16206:Fima C. Klebaner (2005).
15485:Fima C. Klebaner (2005).
15456:Measure Theory (Volume 2)
14903:Fima C. Klebaner (2005).
14792:Fima C. Klebaner (2005).
14635:. SIAM. pp. 14, 15.
14158:. CRC Press. p. 11.
14092:J. F. C. Kingman (1992).
14035:J. F. C. Kingman (1992).
13865:J. F. C. Kingman (1992).
13811:J. F. C. Kingman (1992).
13698:Ubbo F. Wiersema (2008).
13644:Fima C. Klebaner (2005).
13544:Fima C. Klebaner (2005).
13436:Steven E. Shreve (2004).
13041:Fima C. Klebaner (2005).
12960:Fima C. Klebaner (2005).
12763:. Springer. p. 347.
12613:Fima C. Klebaner (2005).
12519:. Springer. p. 133.
12253:Oxford English Dictionary
12013:Oxford English Dictionary
11880:. Springer. p. 120.
11555:Michel Talagrand (2014).
11256:Mikhail Lifshits (2012).
10094:. Wiley. pp. 46, 47.
9500:Mathematical construction
9406:weak law of large numbers
9282:The French mathematician
9130:SchrammâLoewner evolution
8457:is a LĂ©vy process if for
7550:Two stochastic processes
7065:Two stochastic processes
6723:Two stochastic processes
6311:with a probability space
5986:Two stochastic processes
5932:stochastically equivalent
5288:, on a probability space
4053:For a stochastic process
3420:For a stochastic process
1452:, which is a sequence of
1176:{\displaystyle \{X_{t}\}}
904:Oxford English Dictionary
21920:Numerical linear algebra
21231:Doléans-Dade exponential
21061:Progressively measurable
20859:CoxâIngersollâRoss (CIR)
20040:Robert J. Adler (2010).
20033:
19509:Robert J. Adler (2010).
19477:David Applebaum (2004).
18801:Bachelier, Luis (1900).
18760:Bachelier, Luis (1900).
18625:Barry D. Hughes (1995).
17260:10.1016/j.hm.2004.04.001
17057:. Springer. p. 99.
17003:. CRC Press. p. 7.
16792:David Applebaum (2004).
16738:David Applebaum (2004).
16457:Olav Kallenberg (2002).
16400:Sheldon M. Ross (1996).
16244:. Springer. p. 11.
15985:Sheldon M. Ross (1996).
15790:Sheldon M. Ross (1996).
15736:Richard Serfozo (2009).
15682:Petar Todorovic (2012).
15574:Richard F. Bass (2011).
15200:Petar Todorovic (2012).
15071:Olav Kallenberg (2002).
15017:David Applebaum (2004).
14692:. Wiley. pp. 94â96.
14629:Robert J. Adler (2010).
14480:David Applebaum (2004).
13923:Queueing Systems: Theory
13736:. Wiley. pp. 1, 2.
13701:Brownian Motion Calculus
13677:. Springer. p. 47.
13496:. Springer. p. 70.
13463:Olav Kallenberg (2002).
13415:. Springer. p. 61.
13388:. Springer. p. 78.
12811:Olav Kallenberg (2002).
12673:Florescu, Ionut (2014).
12555:Richard F. Bass (2011).
12379:David Stirzaker (2005).
12045:. NG Verlag. p. 5.
11283:Robert J. Adler (2010).
11229:David Applebaum (2004).
10987:Olav Kallenberg (2002).
10792:The Mathematical Gazette
9931:
9034:United States of America
8829:Early probability theory
8717:are all LĂ©vy processes.
8186:Markov chain Monte Carlo
8157:on the integers and the
6805:with the same index set
6432:of probability zero, so
6230:almost surely continuous
6068:with the same index set
5760:, and probability space
5439:{\displaystyle t,s\in T}
5004:probability distribution
4435:onto a finite subset of
4033:probability distribution
3771:For a measurable subset
3675:-valued random variable
2886:, or, particularly when
1210:{\displaystyle \{X(t)\}}
814:. If the state space is
550:settings and countries.
285:Law of total probability
280:Conditional independence
169:Exponential distribution
154:Probability distribution
21659:Finite element analysis
21609:Constraint satisfaction
21451:Mathematical statistics
21441:Large deviations theory
21271:Infinitesimal generator
21132:Maximal ergodic theorem
21051:Piecewise-deterministic
20653:Random dynamical system
20518:Markov additive process
20168:. Courier Corporation.
20082:Pierre Brémaud (2013).
19995:10.1214/lnms/1196285381
19913:Cramer, Harald (1976).
19809:Benjamin Yakir (2013).
19647:Probability and Measure
19590:David Williams (1991).
19563:Bernt Ăksendal (2003).
19184:Pierre Bremaud (2013).
18854:10.1111/1467-9965.00098
18598:Ionut Florescu (2014).
18490:Ionut Florescu (2014).
18077:Bingham, N. H. (2005).
17969:"W. Doeblin, 1915-1940"
17563:Cramer, Harald (1976).
17434:10.1093/biomet/87.1.145
17309:L. E. Maistrov (2014).
17148:L. E. Maistrov (2014).
17051:Volker Schmidt (2014).
16873:Martin Haenggi (2013).
16265:Joseph L. Doob (1990).
16174:Pierre Bremaud (2013).
15904:Emanuel Parzen (2015).
15877:SĂžren Asmussen (2003).
15817:Ionut Florescu (2014).
15515:SĂžren Asmussen (2003).
15326:Joseph L. Doob (1990).
15260:Probability and Measure
15227:Ilya Molchanov (2005).
15105:; Marc Chesney (2009).
15044:Hiroshi Kunita (1997).
14963:Ionut Florescu (2014).
14933:Bernt Ăksendal (2003).
14765:David Williams (1991).
14706:Ionut Florescu (2014).
14686:Joseph L. Doob (1990).
14360:Bernt Ăksendal (2003).
14336:Probability and Measure
14306:Ionut Florescu (2014).
13978:Martin Haenggi (2013).
13157:Ionut Florescu (2014).
12838:Ionut Florescu (2014).
12730:Ibe, Oliver C. (2013).
12643:Ionut Florescu (2014).
12513:Pierre Brémaud (2014).
12429:John A. Gubner (2006).
12349:SĂžren Asmussen (2003).
12258:Oxford University Press
12018:Oxford University Press
11874:Pierre Brémaud (2014).
11815:Probability and Measure
11725:Ionut Florescu (2014).
11475:Adam Bobrowski (2005).
11448:Pierre Brémaud (2014).
11424:Probability and Measure
11175:David Williams (1991).
10744:10.1214/lnms/1196285381
10667:. Courier Corporation.
10503:Baron, Michael (2015).
10357:Oxford University Press
10326:Oxford University Press
10180:Emanuel Parzen (2015).
10088:Joseph L. Doob (1990).
9954:Brownian motion process
9485:characteristic function
9314:kinetic theory of gases
8477:non-negatives numbers,
6272:are indistinguishable.
5978:as their state spaces.
5681:{\displaystyle \Omega }
5407:{\displaystyle s\leq t}
5216:covariance stationarity
5153:{\displaystyle X_{t+h}}
2113:and a measurable space
2061:{\displaystyle \Sigma }
2041:{\displaystyle \sigma }
1949:{\displaystyle \sigma }
1901:{\displaystyle \Omega }
1787:integrable function of
648:as well as branches of
467:, or the movement of a
264:Conditional probability
22370:Statistical data types
22214:Mathematical economics
21888:Multivariable calculus
21771:Differential equations
21614:Constraint programming
21604:Computational geometry
21286:KarhunenâLoĂšve theorem
21221:CameronâMartin formula
21185:BurkholderâDavisâGundy
20580:Variance gamma process
19932:10.1214/aop/1176996025
18517:Oliver C. Ibe (2013).
18265:10.1214/aop/1176991248
18246:Dynkin, E. B. (1989).
18096:10.1239/jap/1110381385
17986:10.1214/aop/1176990329
17945:10.1239/jap/1110381384
17582:10.1214/aop/1176996025
16406:. Wiley. p. 295.
16282:Gilles Pisier (2016).
15958:John Lamperti (1977).
14568:John Lamperti (1977).
14198:10.1215/ijm/1255631584
14065:Roy L. Streit (2010).
13730:Henk C. Tijms (2003).
12586:John Lamperti (1977).
12039:O. B. SheÄnin (2006).
11045:John Lamperti (1977).
9822:
9772:
9752:
9724:
9723:{\displaystyle t\in T}
9694:
9671:
9670:{\displaystyle t\in T}
9641:
9618:
9617:{\displaystyle t\in T}
9588:
9564:
9544:
9515:consistency conditions
9279:
9220:
9084:, with early ideas by
9075:differential equations
9052:
8814:
8779:
8745:
8707:
8660:
8549:
8523:
8471:
8451:
8392:
8294:
8209:
8104:
8069:
8034:
8014:
7986:
7954:
7899:
7746:
7614:
7579:
7536:
7516:
7487:
7334:
7129:
7094:
7051:
6977:
6903:
6851:
6819:
6799:
6757:
6737:
6702:
6682:
6655:
6602:
6543:
6494:
6468:
6426:
6393:
6367:
6347:
6305:
6266:
6246:
6222:
6202:
6180:
6102:
6082:
6062:
6020:
6000:
5968:
5946:but not the converse.
5922:
5862:
5861:{\displaystyle t\in T}
5836:
5816:
5796:
5754:
5734:
5714:
5682:
5662:
5631:
5604:
5584:
5553:
5533:
5532:{\displaystyle t\in T}
5507:
5480:
5460:
5440:
5408:
5382:
5324:
5282:
5200:
5199:{\displaystyle t\in T}
5174:
5154:
5121:
5094:
5093:{\displaystyle t\in T}
5068:
5048:
5023:
4994:
4932:
4912:
4866:
4846:
4819:
4818:{\displaystyle t\in T}
4793:
4754:
4560:
4540:
4492:
4472:
4449:
4429:
4409:
4335:
4277:
4143:
4087:
4067:
4019:
3937:
3915:
3832:
3812:
3785:
3762:
3761:{\displaystyle t\in T}
3736:
3716:
3689:
3669:
3642:
3641:{\displaystyle X^{-1}}
3612:
3611:{\displaystyle \circ }
3592:
3570:
3519:
3505:of stochastic process
3495:
3453:
3415:
3397:
3377:
3357:
3337:
3283:
3243:
3198:
3153:
3115:
3095:
3023:
3003:
2983:
2957:
2900:
2874:
2819:
2793:
2720:
2688:
2657:
2656:{\displaystyle t\in T}
2631:
2611:
2578:
2558:
2522:
2494:
2474:
2450:
2423:
2393:
2367:
2366:{\displaystyle t\in T}
2341:
2288:
2268:
2239:
2238:{\displaystyle t\in T}
2211:
2159:
2139:
2107:
2062:
2042:
2018:
1998:
1974:
1950:
1930:
1902:
1882:
1824:
1801:
1754:nowhere differentiable
1740:
1720:
1695:
1677:
1647:independent increments
1621:
1595:
1569:
1516:
1496:
1470:
1423:
1378:
1340:
1320:
1293:
1260:
1231:
1211:
1177:
1144:
1098:
1038:
1009:
995:when the index set is
934:was used in German by
872:
848:
828:
753:
715:
374:and related fields, a
367:
206:Continuous or discrete
159:Bernoulli distribution
22167:Supersymmetry algebra
22152:Representation theory
22147:Renormalization group
21793:Differential geometry
21674:Optimization software
21646:Mathematical software
21416:Actuarial mathematics
21378:Uniform integrability
21373:Stratonovich integral
21301:LĂ©vyâProkhorov metric
21205:MarcinkiewiczâZygmund
21092:Central limit theorem
20694:Gaussian random field
20523:McKeanâVlasov process
20443:Dyson Brownian motion
20304:GaltonâWatson process
19617:Rick Durrett (2010).
18376:(9): 1041â1044. 2015.
18165:Jean Bertoin (1998).
16687:Jean Bertoin (1998).
16120:Y.A. Rozanov (2012).
15763:Y.A. Rozanov (2012).
15230:Theory of Random Sets
14123:Alexander A. Borovkov
12865:Rick Durrett (2010).
12757:Achim Klenke (2013).
12489:Probability Theory II
12204:Mathematische Annalen
12161:Mathematische Annalen
12127:10.1073/pnas.20.6.376
12100:Doob, Joseph (1934).
10480:John Wiley & Sons
10451:. Athena Scientific.
10443:Bertsekas, Dimitri P.
9823:
9773:
9753:
9725:
9695:
9672:
9642:
9619:
9589:
9565:
9545:
9446:Irénée-Jules Bienaymé
9418:central limit theorem
9318:differential equation
9316:. Einstein derived a
9262:
9226:-dimensional integer
9221:
9042:
8944:mathematical problems
8895:statistical mechanics
8889:Statistical mechanics
8815:
8780:
8746:
8708:
8661:
8550:
8524:
8472:
8452:
8393:
8295:
8210:
8105:
8070:
8035:
8015:
7987:
7955:
7900:
7747:
7615:
7580:
7537:
7517:
7488:
7335:
7130:
7095:
7052:
6978:
6909:, the random vectors
6904:
6852:
6820:
6800:
6758:
6738:
6703:
6683:
6656:
6603:
6544:
6500:and every closed set
6495:
6469:
6427:
6394:
6368:
6348:
6306:
6267:
6247:
6223:
6203:
6181:
6103:
6083:
6063:
6021:
6001:
5969:
5923:
5863:
5837:
5817:
5797:
5755:
5735:
5715:
5683:
5663:
5632:
5630:{\displaystyle X_{t}}
5605:
5585:
5554:
5534:
5508:
5506:{\displaystyle X_{t}}
5481:
5461:
5459:{\displaystyle \leq }
5441:
5409:
5383:
5325:
5283:
5201:
5175:
5155:
5122:
5120:{\displaystyle X_{t}}
5095:
5069:
5049:
5024:
4995:
4933:
4913:
4867:
4847:
4845:{\displaystyle X_{t}}
4820:
4794:
4755:
4561:
4541:
4493:
4473:
4450:
4430:
4410:
4336:
4278:
4144:
4088:
4068:
4020:
3938:
3916:
3833:
3813:
3811:{\displaystyle S^{T}}
3786:
3763:
3742:-valued functions of
3737:
3717:
3715:{\displaystyle S^{T}}
3690:
3670:
3668:{\displaystyle S^{T}}
3643:
3613:
3593:
3571:
3520:
3496:
3454:
3398:
3378:
3358:
3338:
3284:
3244:
3199:
3154:
3116:
3096:
3024:
3004:
2984:
2958:
2901:
2875:
2820:
2794:
2721:
2689:
2658:
2632:
2612:
2579:
2559:
2523:
2495:
2475:
2451:
2449:{\displaystyle S^{T}}
2424:
2422:{\displaystyle S^{T}}
2394:
2368:
2342:
2289:
2269:
2240:
2212:
2160:
2140:
2108:
2063:
2043:
2028:with respect to some
2019:
1999:
1975:
1951:
1931:
1903:
1883:
1825:
1802:
1741:
1721:
1696:
1672:) and without drift (
1667:
1622:
1620:{\displaystyle p=0.5}
1596:
1570:
1517:
1497:
1471:
1424:
1379:
1341:
1321:
1319:{\displaystyle X_{t}}
1294:
1266:even though it is an
1261:
1232:
1212:
1178:
1145:
1099:
1039:
1010:
932:stochastischer ProzeĂ
916:Ladislaus Bortkiewicz
873:
849:
829:
743:
716:
650:mathematical analysis
427:usually defined as a
357:
164:Binomial distribution
22360:Stochastic processes
22219:Mathematical finance
22204:Social choice theory
22119:Algebraic structures
22068:in quantum mechanics
22004:Analytical mechanics
21970:Stochastic processes
21942:Variational calculus
21754:Approximation theory
21679:Statistical software
21491:Time series analysis
21446:Mathematical finance
21331:Reflection principle
20658:Regenerative process
20458:FlemingâViot process
20273:Stochastic processes
20238:at Wikimedia Commons
20236:Stochastic processes
20189:Stochastic Processes
20117:Anders Hald (2005).
20109:Stochastic processes
19515:. SIAM. p. 13.
19456:10.1112/blms/22.1.31
19288:. pp. 331â334.
18842:Mathematical Finance
18773:Serie 3, 17: 21â89.
18571:Anders Hald (2005).
18544:Anders Hald (2005).
18433:Anders Hald (2005).
17833:10.1112/blms/22.1.31
17282:Anders Hald (2005).
17247:Historia Mathematica
16403:Stochastic processes
16268:Stochastic processes
16123:Markov Random Fields
15988:Stochastic processes
15907:Stochastic Processes
15793:Stochastic processes
15766:Markov Random Fields
15577:Stochastic Processes
15332:. Wiley. p. 56.
15329:Stochastic processes
14689:Stochastic processes
12930:Aris Spanos (1999).
12558:Stochastic Processes
11337:Bruce Hajek (2015).
10183:Stochastic Processes
10091:Stochastic processes
9849:Deterministic system
9812:
9762:
9758:a member of the set
9742:
9708:
9684:
9655:
9631:
9602:
9578:
9554:
9534:
9448:. Starting in 1928,
9441:Henry William Watson
9353:Poisson distribution
9210:
9206:For random walks in
9098:Stochastic processes
9049:Stochastic Processes
8942:presented a list of
8867:Pierre-Simon Laplace
8804:
8769:
8735:
8679:
8561:
8533:
8529:, the corresponding
8481:
8461:
8441:
8304:
8245:
8199:
8082:
8044:
8024:
8004:
7964:
7932:
7759:
7628:
7589:
7554:
7526:
7506:
7347:
7143:
7104:
7069:
6987:
6913:
6861:
6833:
6809:
6767:
6747:
6727:
6692:
6665:
6612:
6553:
6504:
6478:
6436:
6403:
6377:
6357:
6315:
6295:
6256:
6236:
6212:
6192:
6118:
6092:
6072:
6030:
6010:
5990:
5958:
5874:
5846:
5826:
5806:
5764:
5744:
5724:
5704:
5672:
5641:
5614:
5594:
5563:
5543:
5517:
5490:
5470:
5450:
5418:
5392:
5334:
5292:
5242:
5184:
5164:
5131:
5104:
5078:
5058:
5038:
5013:
4944:
4922:
4918:, the corresponding
4876:
4856:
4829:
4825:the random variable
4803:
4783:
4572:
4550:
4505:
4482:
4462:
4439:
4428:{\displaystyle \mu }
4419:
4345:
4289:
4155:
4101:
4086:{\displaystyle \mu }
4077:
4057:
3949:
3927:
3844:
3822:
3795:
3775:
3746:
3726:
3699:
3679:
3652:
3622:
3602:
3582:
3535:
3509:
3463:
3424:
3387:
3367:
3347:
3293:
3253:
3208:
3163:
3125:
3105:
3058:
3013:
2993:
2967:
2914:
2890:
2834:
2803:
2750:
2710:
2678:
2641:
2621:
2592:
2568:
2548:
2512:
2484:
2464:
2433:
2406:
2377:
2351:
2298:
2278:
2267:{\displaystyle X(t)}
2249:
2223:
2171:
2149:
2117:
2075:
2052:
2032:
2008:
1988:
1964:
1940:
1916:
1892:
1850:
1814:
1791:
1739:{\displaystyle \mu }
1730:
1719:{\displaystyle \mu }
1710:
1685:
1651:normally distributed
1605:
1579:
1559:
1506:
1480:
1460:
1388:
1350:
1330:
1303:
1292:{\displaystyle X(t)}
1274:
1259:{\displaystyle X(t)}
1241:
1221:
1186:
1154:
1108:
1060:
1019:
999:
862:
838:
818:
705:
290:Law of large numbers
259:Marginal probability
184:Poisson distribution
33:Part of a series on
22194:Operations research
22063:Perturbation theory
21861:Multilinear algebra
21832:Functional analysis
21689:Numerical libraries
21621:Computational logic
21486:Stochastic analysis
21326:Quadratic variation
21321:Prokhorov's theorem
21256:FeynmanâKac formula
20726:Markov random field
20374:Birthâdeath process
19385:2005AmJPh..73..395B
19228:10.1511/2013.101.92
18154:(8): 981â982. 1998.
18037:2009arXiv0909.4213G
17336:John Tabak (2014).
17214:John Tabak (2014).
16589:2006math......9294G
16567:Statistical Science
15709:Ward Whitt (2006).
15434:Gusak et al. (2010)
15404:Ward Whitt (2006).
15314:Gusak et al. (2010)
15143:Kiyosi ItĆ (2006).
14453:Ward Whitt (2006).
13244:; Marc Yor (2013).
12543:Gusak et al. (2010)
12459:Kiyosi ItĆ (2006).
12256:(Online ed.).
12118:1934PNAS...20..376D
12016:(Online ed.).
9910:Stochastic calculus
9870:Gillespie algorithm
9844:Covariance function
9702:increasing function
9596:continuous function
9521:Construction issues
9326:Marian Smoluchowski
9067:stochastic calculus
8903:James Clerk Maxwell
8848:Liber de Ludo Aleae
8548:{\displaystyle n-1}
8190:Bayesian statistics
6636: for all
6577: for all
6399:and there is a set
6152: for all
5009:When the index set
4566:can be written as:
3943:can be written as:
3818:, the pre-image of
3411:Further definitions
3009:to the state space
1982:probability measure
1594:{\displaystyle 1-p}
1495:{\displaystyle 1-p}
958:published in 1888.
666:functional analysis
626:branching processes
463:fluctuating due to
453:mathematical models
425:mathematical object
249:Complementary event
191:Probability measure
179:Pareto distribution
174:Normal distribution
22331:Mathematics portal
22228:Other applications
21952:Probability theory
21935:Validated numerics
21915:Numerical analysis
21808:Geometric analysis
21798:Differential forms
21631:Information theory
21456:Probability theory
21336:Skorokhod integral
21306:Malliavin calculus
20889:Korn-Kreer-Lenssen
20773:Time series models
20736:PitmanâYor process
20144:Stochastic Methods
19902:Notices of the AMS
19785:Probability Theory
19674:Probability Theory
19216:American Scientist
19013:10.1007/BF00328110
18970:10.1007/BF00328110
18816:10.24033/asens.476
18780:10.24033/asens.476
18389:Notices of the AMS
18370:Notices of the AMS
18324:Notices of the AMS
18148:Notices of the AMS
17782:10.1007/BF00328110
17704:10.1007/BF00327232
17661:10.1007/BF00412958
17618:10.1007/BF00327232
16721:Notices of the AMS
15631:Probability Theory
15290:Probability Theory
14879:Probability Theory
14128:Probability Theory
13600:Notices of the AMS
13113:Notices of the AMS
13082:10.1007/BF00328110
12987:Allan Gut (2012).
12216:10.1007/BF01457949
12173:10.1007/BF01449156
11973:Davar Khoshnevisan
11845:Probability Theory
11667:Notices of the AMS
11504:Notices of the AMS
11316:. Academic Press.
10258:Bressloff, Paul C.
9915:Stochastic control
9900:Stationary process
9818:
9798:Anatoliy Skorokhod
9768:
9748:
9720:
9690:
9667:
9637:
9614:
9584:
9560:
9540:
9322:diffusion equation
9280:
9216:
9126:Srinivasa Varadhan
9053:
8971:Aleksandr Khinchin
8810:
8795:random point field
8775:
8741:
8703:
8656:
8545:
8519:
8467:
8447:
8388:
8290:
8205:
8100:
8065:
8030:
8010:
7998:Anatoliy Skorokhod
7982:
7950:
7924:, also written as
7895:
7742:
7610:
7575:
7532:
7512:
7483:
7391: uncorrelated
7330:
7125:
7090:
7047:
6973:
6899:
6847:
6815:
6795:
6753:
6733:
6698:
6678:
6651:
6598:
6539:
6538:
6490:
6464:
6422:
6389:
6363:
6343:
6301:
6262:
6242:
6218:
6198:
6176:
6098:
6078:
6058:
6016:
5996:
5964:
5918:
5858:
5832:
5812:
5792:
5750:
5730:
5710:
5678:
5658:
5627:
5600:
5580:
5549:
5529:
5503:
5476:
5456:
5436:
5404:
5378:
5320:
5278:
5196:
5170:
5150:
5117:
5090:
5064:
5044:
5019:
5002:all have the same
4990:
4928:
4908:
4862:
4842:
4815:
4789:
4772:Stationary process
4750:
4556:
4536:
4488:
4468:
4445:
4425:
4405:
4331:
4273:
4139:
4083:
4063:
4015:
3933:
3911:
3828:
3808:
3781:
3758:
3732:
3712:
3685:
3665:
3638:
3608:
3588:
3566:
3525:is defined as the
3515:
3491:
3449:
3393:
3373:
3353:
3333:
3279:
3239:
3194:
3149:
3111:
3091:
3019:
2999:
2979:
2953:
2896:
2870:
2815:
2789:
2716:
2684:
2673:mathematical space
2653:
2627:
2607:
2574:
2554:
2518:
2490:
2470:
2446:
2419:
2389:
2363:
2337:
2284:
2264:
2235:
2207:
2155:
2135:
2103:
2058:
2038:
2014:
1994:
1970:
1946:
1926:
1898:
1878:
1839:Stochastic process
1820:
1797:
1736:
1716:
1691:
1678:
1617:
1591:
1565:
1553:simple random walk
1512:
1492:
1466:
1419:
1374:
1336:
1316:
1289:
1256:
1227:
1207:
1173:
1140:
1094:
1034:
1005:
985:stochastic process
972:stochastic process
952:stochastic process
936:Aleksandr Khinchin
924:stochastic process
868:
844:
824:
754:
711:
695:mathematical space
618:Gaussian processes
571:mathematical space
563:stochastic process
516:telecommunications
508:information theory
461:electrical current
372:probability theory
368:
300:Boole's inequality
236:Stochastic process
125:Mutual exclusivity
42:Probability theory
22365:Stochastic models
22347:
22346:
22181:Decision sciences
22175:
22174:
22157:Spacetime algebra
21849:Harmonic analysis
21815:Dynamical systems
21759:Clifford analysis
21736:Discrete geometry
21702:
21701:
21522:
21521:
21476:Signal processing
21195:Doob's upcrossing
21190:Doob's martingale
21154:EngelbertâSchmidt
21097:Donsker's theorem
21031:Feller-continuous
20899:RendlemanâBartter
20689:Dirichlet process
20606:Branching process
20575:Telegraph process
20468:Geometric process
20448:Empirical process
20438:Diffusion process
20294:Branching process
20289:Bernoulli process
20234:Media related to
20208:Murray Rosenblatt
20199:978-0-486-79688-8
20175:978-0-486-69387-3
20154:978-3-540-70712-7
20130:978-0-471-72517-6
20095:978-1-4757-3124-8
20074:978-0-387-48116-6
20053:978-0-89871-693-1
20004:978-0-940600-61-4
19891:
19890:
19883:
19822:978-1-118-72062-2
19795:978-1-4471-5201-9
19768:978-0-387-32903-1
19741:978-0-387-48116-6
19711:978-0-387-32903-1
19684:978-1-4471-5201-9
19657:978-81-265-1771-8
19630:978-1-139-49113-6
19603:978-0-521-40605-5
19576:978-3-540-04758-2
19549:978-0-387-32903-1
19522:978-0-89871-693-1
19490:978-0-521-83263-2
19427:978-1-4612-3038-0
19393:10.1119/1.1848117
19357:978-3-540-26312-8
19330:978-1-4471-7262-8
19303:978-0-387-95283-3
19197:978-1-4757-3124-8
19167:978-0-8218-0749-1
19055:978-0-387-21564-8
18638:978-0-19-853788-5
18611:978-1-118-59320-2
18584:978-0-471-72517-6
18557:978-0-471-72517-6
18530:978-1-118-61793-9
18503:978-1-118-59320-2
18476:978-0-444-86806-0
18446:978-0-471-72517-6
18419:978-0-387-21564-8
18354:978-3-642-27933-1
18232:978-1-4832-6322-9
18205:978-1-4684-9305-4
18178:978-0-521-64632-1
18045:10.1214/09-AOP465
17735:Annals of Science
17349:978-0-8160-6873-9
17322:978-1-4832-1863-2
17295:978-0-471-72517-6
17227:978-0-8160-6873-9
17161:978-1-4832-1863-2
17091:978-0-387-21564-8
17064:978-3-319-10064-7
17037:978-0-08-057041-9
17010:978-0-203-49693-0
16983:978-0-19-159124-2
16973:Poisson Processes
16956:978-0-412-21910-8
16913:978-0-387-21564-8
16886:978-1-107-01469-5
16859:978-1-118-65825-3
16832:978-1-118-65825-3
16805:978-0-521-83263-2
16778:978-3-540-68829-7
16751:978-0-521-83263-2
16700:978-0-521-64632-1
16666:978-1-4832-6322-9
16639:978-3-662-11657-9
16551:978-0-19-857222-0
16524:978-1-4684-9305-4
16497:978-1-4684-9305-4
16470:978-0-387-95313-7
16443:978-1-4684-9305-4
16413:978-0-471-12062-9
16386:978-1-4684-9305-4
16359:978-1-4832-6322-9
16325:978-1-4684-9305-4
16295:978-1-316-67946-3
16251:978-1-4612-0949-2
16219:978-1-86094-555-7
16187:978-1-4757-3124-8
16160:978-1-4612-3166-0
16133:978-1-4613-8190-7
16106:978-1-58488-587-0
16079:978-1-118-21052-9
16052:978-0-08-057041-9
16025:978-0-521-73182-9
15998:978-0-471-12062-9
15971:978-3-540-90275-1
15944:978-0-08-057041-9
15917:978-0-486-79688-8
15890:978-0-387-00211-8
15860:978-0-08-057041-9
15830:978-1-118-59320-2
15803:978-0-471-12062-9
15776:978-1-4613-8190-7
15749:978-3-540-89332-5
15722:978-0-387-21748-2
15695:978-1-4613-9742-7
15668:978-0-387-21631-7
15641:978-1-4471-5201-9
15614:978-1-4471-3856-3
15587:978-1-139-50147-7
15560:978-1-118-62596-5
15528:978-0-387-00211-8
15498:978-1-86094-555-7
15466:978-3-540-34514-5
15417:978-0-387-21748-2
15356:978-0-387-21631-7
15300:978-1-4471-5201-9
15270:978-81-265-1771-8
15240:978-1-85233-892-3
15213:978-1-4613-9742-7
15183:978-0-486-69387-3
15156:978-0-8218-3898-3
15118:978-1-85233-376-8
15099:Monique Jeanblanc
15084:978-0-387-95313-7
15057:978-0-521-59925-2
15030:978-0-521-83263-2
15003:978-3-662-06400-9
14976:978-1-118-59320-2
14946:978-3-540-04758-2
14916:978-1-86094-555-7
14889:978-1-4471-5201-9
14862:978-1-107-71749-7
14832:978-1-139-48657-6
14805:978-1-86094-555-7
14778:978-0-521-40605-5
14746:978-0-486-69387-3
14719:978-1-118-59320-2
14669:978-1-118-65825-3
14642:978-0-89871-693-1
14608:978-0-486-69387-3
14581:978-3-540-90275-1
14547:978-1-107-71749-7
14520:978-3-662-06400-9
14493:978-0-521-83263-2
14466:978-0-387-21748-2
14439:978-1-4612-0387-2
14412:978-1-139-48721-4
14373:978-3-540-04758-2
14346:978-81-265-1771-8
14319:978-1-118-59320-2
14292:978-3-540-26312-8
14265:978-1-4612-3166-0
14238:978-3-540-26312-8
14165:978-1-4665-8618-5
14138:978-1-4471-5201-9
14105:978-0-19-159124-2
14095:Poisson Processes
14078:978-1-4419-6923-1
14048:978-0-19-159124-2
14038:Poisson Processes
14021:978-1-118-65825-3
13991:978-1-107-01469-5
13938:978-0-471-49110-1
13927:. Wiley. p.
13905:978-0-08-057041-9
13878:978-0-19-159124-2
13868:Poisson Processes
13851:978-0-387-21564-8
13824:978-0-19-159124-2
13814:Poisson Processes
13797:978-0-12-381416-6
13770:978-0-387-21564-8
13743:978-0-471-49881-0
13711:978-0-470-02171-2
13684:978-1-4612-0949-2
13657:978-1-86094-555-7
13630:978-0-08-057041-9
13584:978-1-4612-0949-2
13557:978-1-86094-555-7
13530:978-1-139-48657-6
13503:978-1-4612-0949-2
13476:978-0-387-95313-7
13449:978-0-387-40101-0
13422:978-1-4612-0949-2
13395:978-1-4612-0949-2
13368:978-1-139-48657-6
13338:978-1-4684-9305-4
13311:978-1-4612-3166-0
13284:978-981-310-165-4
13257:978-3-662-06400-9
13227:978-1-4612-0949-2
13200:978-0-08-057041-9
13170:978-1-118-59320-2
13143:978-0-486-69387-3
13054:978-1-86094-555-7
13027:978-0-19-857222-0
13000:978-1-4614-4708-5
12973:978-1-86094-555-7
12943:978-0-521-42408-0
12878:978-1-139-49113-6
12851:978-1-118-59320-2
12824:978-0-387-95313-7
12797:978-1-139-48876-1
12770:978-1-4471-5362-7
12743:978-1-118-61793-9
12716:978-1-886529-40-3
12686:978-1-118-59320-2
12656:978-1-118-59320-2
12626:978-1-86094-555-7
12599:978-3-540-90275-1
12568:978-1-139-50147-7
12526:978-3-319-09590-5
12499:978-0-387-90262-3
12486:M. LoĂšve (1978).
12472:978-0-8218-3898-3
12442:978-1-139-45717-0
12392:978-0-19-856814-8
12362:978-0-387-00211-8
12326:978-1-107-71749-7
12292:978-1-4899-2837-5
12264:(Subscription or
12079:978-3-89971-812-6
12052:978-3-938417-40-9
12024:(Subscription or
11988:978-0-387-21631-7
11953:978-3-662-12616-5
11923:978-981-310-165-4
11887:978-3-319-09590-5
11855:978-1-4471-5201-9
11825:978-81-265-1771-8
11795:978-1-4612-3166-0
11768:978-0-08-057041-9
11738:978-1-118-59320-2
11700:978-1-107-71749-7
11632:978-0-08-057041-9
11598:978-3-319-08488-6
11583:Paul C. Bressloff
11568:978-3-642-54075-2
11541:978-3-03719-072-2
11488:978-0-521-83166-6
11461:978-3-319-09590-5
11434:978-81-265-1771-8
11407:978-0-387-21337-8
11380:978-0-89871-425-8
11350:978-1-316-24124-0
11323:978-0-08-057041-9
11296:978-0-89871-693-1
11269:978-3-642-24939-6
11242:978-0-521-83263-2
11215:978-1-107-71749-7
11188:978-0-521-40605-5
11161:978-1-139-48876-1
11134:978-0-387-48116-6
11092:978-1-107-60655-5
11058:978-3-540-90275-1
11000:978-0-387-95313-7
10962:978-3-540-26312-8
10935:978-0-387-87862-1
10854:978-1-4612-3166-0
10753:978-0-940600-61-4
10674:978-0-486-69387-3
10647:978-0-387-40101-0
10616:978-3-540-26653-2
10582:978-0-387-95016-7
10551:978-1-60198-264-3
10520:978-1-4987-6060-7
10489:978-1-118-58577-1
10428:978-0-8194-2513-3
10397:978-3-319-00327-6
10366:978-0-19-923507-0
10335:978-0-19-852525-7
10304:978-0-08-047536-3
10273:978-3-319-08488-6
10243:978-0-486-69387-3
10193:978-0-486-79688-8
10157:978-1-4684-9305-4
10125:978-1-107-71749-7
9950:Brownian movement
9920:Stochastic parrot
9905:Statistical model
9821:{\displaystyle n}
9802:Andrei Kolmogorov
9771:{\displaystyle T}
9751:{\displaystyle t}
9734:where the symbol
9693:{\displaystyle X}
9640:{\displaystyle X}
9587:{\displaystyle X}
9563:{\displaystyle T}
9543:{\displaystyle X}
9456:Andrei Kolmogorov
9433:Tatyana Ehrenfest
9420:for such chains.
9414:Alexander Pushkin
9380:Ernest Rutherford
9219:{\displaystyle n}
9201:Abraham de Moivre
9164:Bernoulli process
9140:being awarded to
9122:Monroe D. Donsker
8975:Andrei Kolmogorov
8920:Brownian movement
8883:Pafnuty Chebyshev
8871:Abraham de Moivre
8813:{\displaystyle n}
8778:{\displaystyle n}
8744:{\displaystyle n}
8470:{\displaystyle n}
8450:{\displaystyle X}
8241:random variables
8208:{\displaystyle n}
8068:{\displaystyle D}
8033:{\displaystyle D}
8013:{\displaystyle D}
7804:
7737:
7535:{\displaystyle Y}
7515:{\displaystyle X}
7392:
7057:are independent.
6818:{\displaystyle T}
6756:{\displaystyle Y}
6736:{\displaystyle X}
6701:{\displaystyle n}
6637:
6578:
6549:, the two events
6366:{\displaystyle T}
6304:{\displaystyle X}
6265:{\displaystyle Y}
6245:{\displaystyle X}
6221:{\displaystyle Y}
6201:{\displaystyle X}
6153:
6112:if the following
6110:indistinguishable
6101:{\displaystyle S}
6081:{\displaystyle T}
6019:{\displaystyle Y}
5999:{\displaystyle X}
5982:Indistinguishable
5967:{\displaystyle n}
5835:{\displaystyle Y}
5815:{\displaystyle Y}
5753:{\displaystyle S}
5733:{\displaystyle T}
5713:{\displaystyle X}
5603:{\displaystyle t}
5552:{\displaystyle t}
5479:{\displaystyle T}
5173:{\displaystyle h}
5067:{\displaystyle X}
5047:{\displaystyle X}
5022:{\displaystyle T}
4938:random variables
4931:{\displaystyle n}
4872:index set values
4865:{\displaystyle n}
4792:{\displaystyle X}
4559:{\displaystyle X}
4491:{\displaystyle n}
4471:{\displaystyle C}
4448:{\displaystyle T}
4066:{\displaystyle X}
3936:{\displaystyle X}
3831:{\displaystyle X}
3784:{\displaystyle B}
3735:{\displaystyle S}
3688:{\displaystyle X}
3591:{\displaystyle P}
3518:{\displaystyle X}
3396:{\displaystyle n}
3376:{\displaystyle S}
3356:{\displaystyle S}
3289:, the difference
3114:{\displaystyle S}
3022:{\displaystyle S}
3002:{\displaystyle T}
2899:{\displaystyle T}
2719:{\displaystyle n}
2687:{\displaystyle S}
2630:{\displaystyle n}
2577:{\displaystyle T}
2557:{\displaystyle T}
2521:{\displaystyle T}
2493:{\displaystyle S}
2473:{\displaystyle T}
2287:{\displaystyle t}
2158:{\displaystyle S}
2017:{\displaystyle S}
1997:{\displaystyle T}
1973:{\displaystyle P}
1845:probability space
1823:{\displaystyle n}
1800:{\displaystyle t}
1758:Donsker's theorem
1694:{\displaystyle n}
1659:Brownian movement
1568:{\displaystyle p}
1515:{\displaystyle p}
1469:{\displaystyle p}
1450:Bernoulli process
1444:Bernoulli process
1438:Bernoulli process
1339:{\displaystyle t}
1230:{\displaystyle X}
1008:{\displaystyle n}
956:Francis Edgeworth
871:{\displaystyle n}
847:{\displaystyle n}
827:{\displaystyle n}
714:{\displaystyle n}
622:renewal processes
620:, random fields,
520:financial markets
500:signal processing
437:probability space
352:
351:
254:Joint probability
201:Bernoulli process
100:Probability space
16:(Redirected from
22377:
22132:Feynman integral
22115:
22114:
22075:Potential theory
21964:random variables
21854:Fourier analysis
21837:Operator algebra
21764:Clifford algebra
21716:Computer algebra
21642:
21641:
21549:
21542:
21535:
21526:
21525:
21496:Machine learning
21383:Usual hypotheses
21266:Girsanov theorem
21251:Dynkin's formula
21016:Continuous paths
20924:Actuarial models
20864:GarmanâKohlhagen
20834:BlackâKarasinski
20829:BlackâDermanâToy
20816:Financial models
20682:Fields and other
20611:Gaussian process
20560:Sigma-martingale
20364:Additive process
20266:
20259:
20252:
20243:
20242:
20233:
20219:
20217:
20214:Random Processes
20203:
20179:
20158:
20139:Crispin Gardiner
20134:
20113:
20099:
20078:
20057:
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20016:
19981:
19944:
19934:
19909:
19908:(11): 1336â1347.
19886:
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19875:
19872:
19866:
19861:Please read the
19857:may need cleanup
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18395:(11): 1341â1344.
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15374:
15369:Lapidoth, Amos,
15367:
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14110:
14109:
14089:
14083:
14082:
14062:
14053:
14052:
14032:
14026:
14025:
14005:
13996:
13995:
13975:
13966:
13965:
13959:
13956:Random Processes
13949:
13943:
13942:
13926:
13916:
13910:
13909:
13889:
13883:
13882:
13862:
13856:
13855:
13835:
13829:
13828:
13808:
13802:
13801:
13781:
13775:
13774:
13754:
13748:
13747:
13727:
13716:
13715:
13695:
13689:
13688:
13668:
13662:
13661:
13641:
13635:
13634:
13614:
13608:
13607:
13595:
13589:
13588:
13568:
13562:
13561:
13541:
13535:
13534:
13514:
13508:
13507:
13487:
13481:
13480:
13460:
13454:
13453:
13433:
13427:
13426:
13406:
13400:
13399:
13379:
13373:
13372:
13352:
13343:
13342:
13322:
13316:
13315:
13295:
13289:
13288:
13268:
13262:
13261:
13238:
13232:
13231:
13211:
13205:
13204:
13184:
13175:
13174:
13154:
13148:
13147:
13127:
13121:
13120:
13108:
13102:
13101:
13065:
13059:
13058:
13038:
13032:
13031:
13011:
13005:
13004:
12984:
12978:
12977:
12957:
12948:
12947:
12927:
12921:
12920:
12894:
12883:
12882:
12862:
12856:
12855:
12835:
12829:
12828:
12808:
12802:
12801:
12781:
12775:
12774:
12754:
12748:
12747:
12727:
12721:
12720:
12700:
12691:
12690:
12670:
12661:
12660:
12640:
12631:
12630:
12610:
12604:
12603:
12582:
12573:
12572:
12552:
12546:
12540:
12531:
12530:
12510:
12504:
12503:
12483:
12477:
12476:
12456:
12447:
12446:
12426:
12420:
12419:
12413:
12410:Random Processes
12403:
12397:
12396:
12376:
12367:
12366:
12346:
12331:
12330:
12310:
12297:
12296:
12276:
12270:
12269:
12261:
12249:
12242:
12236:
12235:
12199:
12193:
12192:
12156:
12150:
12149:
12139:
12129:
12097:
12084:
12083:
12063:
12057:
12056:
12036:
12030:
12029:
12021:
12009:
12002:
11993:
11992:
11969:
11958:
11957:
11937:
11928:
11927:
11907:
11892:
11891:
11871:
11860:
11859:
11839:
11830:
11829:
11809:
11800:
11799:
11779:
11773:
11772:
11752:
11743:
11742:
11722:
11705:
11704:
11684:
11675:
11674:
11662:
11637:
11636:
11616:
11603:
11602:
11579:
11573:
11572:
11552:
11546:
11545:
11521:
11512:
11511:
11510:(11): 1336â1347.
11499:
11493:
11492:
11472:
11466:
11465:
11445:
11439:
11438:
11418:
11412:
11411:
11391:
11385:
11384:
11364:
11355:
11354:
11334:
11328:
11327:
11307:
11301:
11300:
11280:
11274:
11273:
11253:
11247:
11246:
11226:
11220:
11219:
11199:
11193:
11192:
11172:
11166:
11165:
11145:
11139:
11138:
11118:
11097:
11096:
11076:
11063:
11062:
11042:
11005:
11004:
10984:
10967:
10966:
10946:
10940:
10939:
10915:
10909:
10908:
10872:
10859:
10858:
10838:
10832:
10831:
10798:(500): 197â210.
10787:
10766:
10765:
10737:
10721:
10698:
10697:
10695:
10692:Random Processes
10685:
10679:
10678:
10658:
10652:
10651:
10627:
10621:
10620:
10596:
10587:
10586:
10562:
10556:
10555:
10531:
10525:
10524:
10509:(2nd ed.).
10500:
10494:
10493:
10469:
10463:
10462:
10439:
10433:
10432:
10408:
10402:
10401:
10377:
10371:
10370:
10346:
10340:
10339:
10315:
10309:
10308:
10284:
10278:
10277:
10254:
10248:
10247:
10227:
10198:
10197:
10177:
10162:
10161:
10141:
10130:
10129:
10109:
10096:
10095:
10085:
10051:
10048:
10042:
10039:
10033:
10030:
10024:
10021:
10015:
10012:
10006:
10003:
9997:
9994:
9988:
9985:
9979:
9975:
9969:
9942:
9827:
9825:
9824:
9819:
9777:
9775:
9774:
9769:
9757:
9755:
9754:
9749:
9729:
9727:
9726:
9721:
9699:
9697:
9696:
9691:
9676:
9674:
9673:
9668:
9649:bounded function
9646:
9644:
9643:
9638:
9623:
9621:
9620:
9615:
9593:
9591:
9590:
9585:
9569:
9567:
9566:
9561:
9549:
9547:
9546:
9541:
9490:Bruno de Finetti
9395:Markov processes
9254:Kalman filtering
9225:
9223:
9222:
9217:
9189:coined the term
9106:Sergei Bernstein
9102:Paul-André Meyer
9082:potential theory
9018:Wolfgang Doeblin
8967:Sergei Bernstein
8908:Ludwig Boltzmann
8852:Gerolamo Cardano
8844:gambling problem
8819:
8817:
8816:
8811:
8784:
8782:
8781:
8776:
8750:
8748:
8747:
8742:
8712:
8710:
8709:
8704:
8665:
8663:
8662:
8657:
8652:
8651:
8650:
8649:
8626:
8625:
8624:
8623:
8600:
8599:
8598:
8597:
8580:
8579:
8578:
8577:
8554:
8552:
8551:
8546:
8528:
8526:
8525:
8520:
8518:
8517:
8499:
8498:
8476:
8474:
8473:
8468:
8456:
8454:
8453:
8448:
8397:
8395:
8394:
8389:
8381:
8380:
8368:
8367:
8355:
8354:
8342:
8341:
8329:
8328:
8316:
8315:
8299:
8297:
8296:
8291:
8283:
8282:
8270:
8269:
8257:
8256:
8214:
8212:
8211:
8206:
8131:Further examples
8109:
8107:
8106:
8103:{\displaystyle }
8101:
8074:
8072:
8071:
8066:
8039:
8037:
8036:
8031:
8019:
8017:
8016:
8011:
7991:
7989:
7988:
7983:
7959:
7957:
7956:
7953:{\displaystyle }
7951:
7904:
7902:
7901:
7896:
7894:
7893:
7881:
7880:
7858:
7857:
7845:
7844:
7829:
7828:
7827:
7822:
7805:
7803: orthogonal
7802:
7800:
7796:
7795:
7779:
7775:
7774:
7751:
7749:
7748:
7743:
7738:
7733:
7729:
7728:
7712:
7707:
7706:
7676:
7675:
7663:
7662:
7647:
7646:
7645:
7640:
7619:
7617:
7616:
7611:
7609:
7605:
7604:
7584:
7582:
7581:
7576:
7574:
7570:
7569:
7541:
7539:
7538:
7533:
7521:
7519:
7518:
7513:
7492:
7490:
7489:
7484:
7482:
7481:
7469:
7468:
7446:
7445:
7433:
7432:
7417:
7416:
7415:
7410:
7393:
7390:
7388:
7384:
7383:
7367:
7363:
7362:
7339:
7337:
7336:
7331:
7329:
7325:
7324:
7320:
7316:
7315:
7303:
7302:
7287:
7286:
7266:
7262:
7258:
7257:
7245:
7244:
7229:
7228:
7191:
7190:
7178:
7177:
7162:
7161:
7160:
7155:
7134:
7132:
7131:
7126:
7124:
7120:
7119:
7099:
7097:
7096:
7091:
7089:
7085:
7084:
7061:Uncorrelatedness
7056:
7054:
7053:
7048:
7046:
7042:
7038:
7037:
7010:
7009:
6982:
6980:
6979:
6974:
6972:
6968:
6964:
6963:
6936:
6935:
6908:
6906:
6905:
6900:
6892:
6891:
6873:
6872:
6856:
6854:
6853:
6848:
6846:
6824:
6822:
6821:
6816:
6804:
6802:
6801:
6796:
6785:
6784:
6762:
6760:
6759:
6754:
6742:
6740:
6739:
6734:
6707:
6705:
6704:
6699:
6687:
6685:
6684:
6679:
6677:
6676:
6660:
6658:
6657:
6652:
6638:
6635:
6627:
6626:
6607:
6605:
6604:
6599:
6579:
6576:
6568:
6567:
6548:
6546:
6545:
6540:
6499:
6497:
6496:
6491:
6473:
6471:
6470:
6465:
6454:
6453:
6431:
6429:
6428:
6423:
6415:
6414:
6398:
6396:
6395:
6390:
6372:
6370:
6369:
6364:
6352:
6350:
6349:
6344:
6333:
6332:
6310:
6308:
6307:
6302:
6271:
6269:
6268:
6263:
6251:
6249:
6248:
6243:
6227:
6225:
6224:
6219:
6207:
6205:
6204:
6199:
6185:
6183:
6182:
6177:
6154:
6151:
6149:
6148:
6136:
6135:
6107:
6105:
6104:
6099:
6087:
6085:
6084:
6079:
6067:
6065:
6064:
6059:
6048:
6047:
6025:
6023:
6022:
6017:
6005:
6003:
6002:
5997:
5973:
5971:
5970:
5965:
5927:
5925:
5924:
5919:
5905:
5904:
5892:
5891:
5867:
5865:
5864:
5859:
5841:
5839:
5838:
5833:
5821:
5819:
5818:
5813:
5801:
5799:
5798:
5793:
5782:
5781:
5759:
5757:
5756:
5751:
5739:
5737:
5736:
5731:
5719:
5717:
5716:
5711:
5687:
5685:
5684:
5679:
5667:
5665:
5664:
5659:
5657:
5656:
5651:
5650:
5636:
5634:
5633:
5628:
5626:
5625:
5609:
5607:
5606:
5601:
5590:is that as time
5589:
5587:
5586:
5581:
5579:
5578:
5573:
5572:
5558:
5556:
5555:
5550:
5538:
5536:
5535:
5530:
5512:
5510:
5509:
5504:
5502:
5501:
5485:
5483:
5482:
5477:
5465:
5463:
5462:
5457:
5445:
5443:
5442:
5437:
5413:
5411:
5410:
5405:
5387:
5385:
5384:
5379:
5377:
5376:
5367:
5366:
5361:
5360:
5350:
5349:
5344:
5343:
5329:
5327:
5326:
5321:
5310:
5309:
5287:
5285:
5284:
5279:
5277:
5276:
5261:
5260:
5255:
5254:
5205:
5203:
5202:
5197:
5179:
5177:
5176:
5171:
5159:
5157:
5156:
5151:
5149:
5148:
5126:
5124:
5123:
5118:
5116:
5115:
5099:
5097:
5096:
5091:
5073:
5071:
5070:
5065:
5053:
5051:
5050:
5045:
5028:
5026:
5025:
5020:
4999:
4997:
4996:
4991:
4986:
4985:
4984:
4983:
4963:
4962:
4961:
4960:
4937:
4935:
4934:
4929:
4917:
4915:
4914:
4909:
4907:
4906:
4888:
4887:
4871:
4869:
4868:
4863:
4851:
4849:
4848:
4843:
4841:
4840:
4824:
4822:
4821:
4816:
4798:
4796:
4795:
4790:
4759:
4757:
4756:
4751:
4746:
4745:
4739:
4738:
4726:
4725:
4710:
4709:
4708:
4707:
4675:
4674:
4673:
4672:
4658:
4657:
4639:
4638:
4632:
4631:
4610:
4609:
4608:
4607:
4589:
4588:
4565:
4563:
4562:
4557:
4545:
4543:
4542:
4537:
4517:
4516:
4497:
4495:
4494:
4489:
4477:
4475:
4474:
4469:
4454:
4452:
4451:
4446:
4434:
4432:
4431:
4426:
4414:
4412:
4411:
4406:
4398:
4397:
4396:
4368:
4367:
4366:
4340:
4338:
4337:
4332:
4330:
4329:
4328:
4327:
4306:
4305:
4282:
4280:
4279:
4274:
4269:
4268:
4253:
4252:
4251:
4223:
4222:
4221:
4193:
4192:
4191:
4190:
4172:
4171:
4148:
4146:
4145:
4140:
4132:
4131:
4113:
4112:
4092:
4090:
4089:
4084:
4072:
4070:
4069:
4064:
4024:
4022:
4021:
4016:
3942:
3940:
3939:
3934:
3923:so the law of a
3920:
3918:
3917:
3912:
3859:
3858:
3837:
3835:
3834:
3829:
3817:
3815:
3814:
3809:
3807:
3806:
3790:
3788:
3787:
3782:
3767:
3765:
3764:
3759:
3741:
3739:
3738:
3733:
3721:
3719:
3718:
3713:
3711:
3710:
3694:
3692:
3691:
3686:
3674:
3672:
3671:
3666:
3664:
3663:
3647:
3645:
3644:
3639:
3637:
3636:
3617:
3615:
3614:
3609:
3597:
3595:
3594:
3589:
3575:
3573:
3572:
3567:
3562:
3561:
3524:
3522:
3521:
3516:
3500:
3498:
3497:
3492:
3481:
3480:
3458:
3456:
3455:
3450:
3448:
3447:
3402:
3400:
3399:
3394:
3382:
3380:
3379:
3374:
3362:
3360:
3359:
3354:
3342:
3340:
3339:
3334:
3332:
3331:
3330:
3329:
3312:
3311:
3310:
3309:
3288:
3286:
3285:
3280:
3278:
3277:
3265:
3264:
3248:
3246:
3245:
3240:
3220:
3219:
3203:
3201:
3200:
3195:
3175:
3174:
3158:
3156:
3155:
3150:
3120:
3118:
3117:
3112:
3100:
3098:
3097:
3092:
3028:
3026:
3025:
3020:
3008:
3006:
3005:
3000:
2988:
2986:
2985:
2980:
2962:
2960:
2959:
2954:
2905:
2903:
2902:
2897:
2879:
2877:
2876:
2871:
2824:
2822:
2821:
2816:
2798:
2796:
2795:
2790:
2728:Euclidean spaces
2725:
2723:
2722:
2717:
2693:
2691:
2690:
2685:
2662:
2660:
2659:
2654:
2636:
2634:
2633:
2628:
2616:
2614:
2613:
2608:
2606:
2605:
2600:
2583:
2581:
2580:
2575:
2563:
2561:
2560:
2555:
2527:
2525:
2524:
2519:
2499:
2497:
2496:
2491:
2479:
2477:
2476:
2471:
2455:
2453:
2452:
2447:
2445:
2444:
2428:
2426:
2425:
2420:
2418:
2417:
2398:
2396:
2395:
2390:
2372:
2370:
2369:
2364:
2346:
2344:
2343:
2338:
2293:
2291:
2290:
2285:
2273:
2271:
2270:
2265:
2244:
2242:
2241:
2236:
2216:
2214:
2213:
2208:
2164:
2162:
2161:
2156:
2144:
2142:
2141:
2136:
2112:
2110:
2109:
2104:
2093:
2092:
2067:
2065:
2064:
2059:
2047:
2045:
2044:
2039:
2024:, which must be
2023:
2021:
2020:
2015:
2003:
2001:
2000:
1995:
1979:
1977:
1976:
1971:
1955:
1953:
1952:
1947:
1935:
1933:
1932:
1927:
1925:
1924:
1907:
1905:
1904:
1899:
1887:
1885:
1884:
1879:
1868:
1867:
1829:
1827:
1826:
1821:
1806:
1804:
1803:
1798:
1745:
1743:
1742:
1737:
1725:
1723:
1722:
1717:
1700:
1698:
1697:
1692:
1675:
1671:
1626:
1624:
1623:
1618:
1600:
1598:
1597:
1592:
1574:
1572:
1571:
1566:
1521:
1519:
1518:
1513:
1501:
1499:
1498:
1493:
1475:
1473:
1472:
1467:
1428:
1426:
1425:
1420:
1403:
1402:
1383:
1381:
1380:
1375:
1345:
1343:
1342:
1337:
1325:
1323:
1322:
1317:
1315:
1314:
1298:
1296:
1295:
1290:
1265:
1263:
1262:
1257:
1236:
1234:
1233:
1228:
1216:
1214:
1213:
1208:
1182:
1180:
1179:
1174:
1169:
1168:
1149:
1147:
1146:
1141:
1139:
1138:
1123:
1122:
1103:
1101:
1100:
1095:
1093:
1092:
1043:
1041:
1040:
1035:
1033:
1032:
1027:
1014:
1012:
1011:
1006:
877:
875:
874:
869:
853:
851:
850:
845:
833:
831:
830:
825:
720:
718:
717:
712:
662:Fourier analysis
610:Markov processes
512:computer science
496:image processing
433:random variables
418:
417:
414:
413:
410:
407:
404:
401:
398:
395:
392:
389:
386:
344:
337:
330:
120:Elementary event
52:
30:
29:
21:
22385:
22384:
22380:
22379:
22378:
22376:
22375:
22374:
22350:
22349:
22348:
22343:
22315:
22277:
22261:
22223:
22171:
22137:Poisson algebra
22113:
21995:
21988:
21946:
21842:Operator theory
21740:
21698:
21664:Tensor software
21640:
21589:Automata theory
21558:
21553:
21523:
21518:
21500:
21461:Queueing theory
21404:
21346:Skorokhod space
21209:
21200:KunitaâWatanabe
21171:
21137:Sanov's theorem
21107:Ergodic theorem
21080:
21076:Time-reversible
20994:
20957:Queueing models
20951:
20947:SparreâAnderson
20937:CramĂ©râLundberg
20918:
20904:SABR volatility
20810:
20767:
20719:Boolean network
20677:
20663:Renewal process
20594:
20543:Non-homogeneous
20533:Poisson process
20423:Contact process
20386:Brownian motion
20356:Continuous time
20350:
20344:Maximal entropy
20275:
20270:
20226:
20200:
20176:
20155:
20131:
20096:
20075:
20054:
20036:
20005:
19896:
19887:
19876:
19870:
19867:
19860:
19853:Further reading
19847:
19843:
19836:
19834:Further reading
19831:
19830:
19823:
19807:
19803:
19796:
19780:
19776:
19769:
19753:
19749:
19742:
19726:
19719:
19712:
19696:
19692:
19685:
19669:
19665:
19658:
19642:
19638:
19631:
19615:
19611:
19604:
19588:
19584:
19577:
19561:
19557:
19550:
19534:
19530:
19523:
19507:
19498:
19491:
19475:
19471:
19439:
19435:
19428:
19412:
19408:
19369:
19365:
19358:
19342:
19338:
19331:
19315:
19311:
19304:
19282:
19278:
19255:10.2307/1403518
19239:
19235:
19212:
19205:
19198:
19182:
19175:
19168:
19148:
19139:
19108:
19104:
19089:
19067:
19063:
19056:
19040:
19036:
18997:
18993:
18954:
18950:
18942:
18903:
18897:
18884:
18876:
18837:
18831:
18824:
18799:
18795:
18787:
18764:
18758:
18754:
18731:10.2307/1402616
18715:
18708:
18685:10.2307/1403034
18669:
18665:
18650:
18646:
18639:
18623:
18619:
18612:
18596:
18592:
18585:
18569:
18565:
18558:
18542:
18538:
18531:
18515:
18511:
18504:
18488:
18484:
18477:
18461:
18454:
18447:
18431:
18427:
18420:
18404:
18400:
18385:
18381:
18367:
18366:
18362:
18355:
18339:
18335:
18320:
18316:
18285:
18281:
18244:
18240:
18233:
18217:
18213:
18206:
18190:
18186:
18179:
18163:
18159:
18145:
18144:
18140:
18125:
18112:
18075:
18068:
18011:
18002:
17965:
17961:
17924:
17917:
17910:
17888:
17881:
17874:
17852:
17848:
17816:
17805:
17766:
17762:
17731:
17727:
17688:
17684:
17645:
17641:
17602:
17598:
17561:
17538:
17515:10.2307/2974673
17499:
17495:
17456:
17449:
17418:
17403:
17380:10.2307/2589523
17364:
17357:
17350:
17334:
17330:
17323:
17307:
17303:
17296:
17280:
17276:
17239:
17235:
17228:
17212:
17208:
17201:
17176:
17169:
17162:
17146:
17142:
17119:10.2307/2333419
17103:
17099:
17092:
17076:
17072:
17065:
17049:
17045:
17038:
17022:
17018:
17011:
16995:
16991:
16984:
16968:
16964:
16957:
16942:Point Processes
16930:
16921:
16914:
16898:
16894:
16887:
16871:
16867:
16860:
16844:
16840:
16833:
16817:
16813:
16806:
16790:
16786:
16779:
16763:
16759:
16752:
16736:
16732:
16717:
16708:
16701:
16685:
16674:
16667:
16651:
16647:
16640:
16624:
16620:
16573:(2): 292, 293.
16563:
16559:
16552:
16536:
16532:
16525:
16509:
16505:
16498:
16482:
16478:
16471:
16455:
16451:
16444:
16428:
16421:
16414:
16398:
16394:
16387:
16371:
16367:
16360:
16340:
16333:
16326:
16310:
16303:
16296:
16280:
16276:
16263:
16259:
16252:
16236:
16227:
16220:
16204:
16195:
16188:
16172:
16168:
16161:
16145:
16141:
16134:
16118:
16114:
16107:
16091:
16087:
16080:
16064:
16060:
16053:
16037:
16033:
16026:
16010:
16006:
15999:
15983:
15979:
15972:
15956:
15952:
15945:
15929:
15925:
15918:
15902:
15898:
15891:
15875:
15868:
15861:
15845:
15838:
15831:
15815:
15811:
15804:
15788:
15784:
15777:
15761:
15757:
15750:
15734:
15730:
15723:
15707:
15703:
15696:
15680:
15676:
15669:
15653:
15649:
15642:
15626:
15622:
15615:
15599:
15595:
15588:
15572:
15568:
15561:
15545:
15536:
15529:
15513:
15506:
15499:
15483:
15474:
15467:
15451:
15440:
15432:
15425:
15418:
15402:
15391:
15386:
15377:
15368:
15364:
15357:
15341:
15337:
15324:
15320:
15312:
15308:
15301:
15285:
15278:
15271:
15255:
15248:
15241:
15225:
15221:
15214:
15198:
15191:
15184:
15168:
15164:
15157:
15141:
15126:
15119:
15096:
15092:
15085:
15069:
15065:
15058:
15042:
15038:
15031:
15015:
15011:
15004:
14988:
14984:
14977:
14961:
14954:
14947:
14931:
14924:
14917:
14901:
14897:
14890:
14874:
14870:
14863:
14847:
14840:
14833:
14822:Brownian Motion
14817:
14813:
14806:
14790:
14786:
14779:
14763:
14754:
14747:
14731:
14727:
14720:
14704:
14697:
14684:
14677:
14670:
14654:
14650:
14643:
14627:
14616:
14609:
14593:
14589:
14582:
14566:
14555:
14548:
14532:
14528:
14521:
14505:
14501:
14494:
14478:
14474:
14467:
14451:
14447:
14440:
14424:
14420:
14413:
14394:
14381:
14374:
14358:
14354:
14347:
14331:
14327:
14320:
14304:
14300:
14293:
14277:
14273:
14266:
14250:
14246:
14239:
14223:
14214:
14177:
14173:
14166:
14150:
14146:
14139:
14120:
14113:
14106:
14090:
14086:
14079:
14063:
14056:
14049:
14033:
14029:
14022:
14006:
13999:
13992:
13976:
13969:
13950:
13946:
13939:
13917:
13913:
13906:
13890:
13886:
13879:
13863:
13859:
13852:
13836:
13832:
13825:
13809:
13805:
13798:
13782:
13778:
13771:
13755:
13751:
13744:
13728:
13719:
13712:
13696:
13692:
13685:
13669:
13665:
13658:
13642:
13638:
13631:
13615:
13611:
13596:
13592:
13585:
13569:
13565:
13558:
13542:
13538:
13531:
13520:Brownian Motion
13515:
13511:
13504:
13488:
13484:
13477:
13461:
13457:
13450:
13434:
13430:
13423:
13407:
13403:
13396:
13380:
13376:
13369:
13358:Brownian Motion
13353:
13346:
13339:
13323:
13319:
13312:
13296:
13292:
13285:
13269:
13265:
13258:
13239:
13235:
13228:
13212:
13208:
13201:
13185:
13178:
13171:
13155:
13151:
13144:
13128:
13124:
13109:
13105:
13066:
13062:
13055:
13039:
13035:
13028:
13012:
13008:
13001:
12985:
12981:
12974:
12958:
12951:
12944:
12928:
12924:
12917:
12895:
12886:
12879:
12863:
12859:
12852:
12836:
12832:
12825:
12809:
12805:
12798:
12782:
12778:
12771:
12755:
12751:
12744:
12728:
12724:
12717:
12701:
12694:
12687:
12671:
12664:
12657:
12641:
12634:
12627:
12611:
12607:
12600:
12583:
12576:
12569:
12553:
12549:
12541:
12534:
12527:
12511:
12507:
12500:
12484:
12480:
12473:
12457:
12450:
12443:
12427:
12423:
12404:
12400:
12393:
12377:
12370:
12363:
12347:
12334:
12327:
12311:
12300:
12293:
12277:
12273:
12263:
12243:
12239:
12200:
12196:
12157:
12153:
12098:
12087:
12080:
12064:
12060:
12053:
12037:
12033:
12023:
12003:
11996:
11989:
11970:
11961:
11954:
11938:
11931:
11924:
11908:
11895:
11888:
11872:
11863:
11856:
11840:
11833:
11826:
11810:
11803:
11796:
11780:
11776:
11769:
11753:
11746:
11739:
11723:
11708:
11701:
11685:
11678:
11663:
11640:
11633:
11617:
11606:
11599:
11580:
11576:
11569:
11553:
11549:
11542:
11522:
11515:
11500:
11496:
11489:
11473:
11469:
11462:
11446:
11442:
11435:
11419:
11415:
11408:
11392:
11388:
11381:
11365:
11358:
11351:
11335:
11331:
11324:
11308:
11304:
11297:
11281:
11277:
11270:
11254:
11250:
11243:
11227:
11223:
11216:
11200:
11196:
11189:
11173:
11169:
11162:
11146:
11142:
11135:
11119:
11100:
11093:
11077:
11066:
11059:
11043:
11008:
11001:
10985:
10970:
10963:
10947:
10943:
10936:
10920:Mishura, Yuliya
10916:
10912:
10873:
10862:
10855:
10839:
10835:
10804:10.2307/3621649
10788:
10769:
10754:
10722:
10701:
10686:
10682:
10675:
10659:
10655:
10648:
10628:
10624:
10617:
10597:
10590:
10583:
10563:
10559:
10552:
10532:
10528:
10521:
10513:. p. 131.
10501:
10497:
10490:
10470:
10466:
10459:
10440:
10436:
10429:
10409:
10405:
10398:
10378:
10374:
10367:
10347:
10343:
10336:
10316:
10312:
10305:
10285:
10281:
10274:
10255:
10251:
10244:
10228:
10201:
10194:
10178:
10165:
10158:
10142:
10133:
10126:
10110:
10099:
10086:
10065:
10060:
10055:
10054:
10049:
10045:
10040:
10036:
10031:
10027:
10022:
10018:
10013:
10009:
10004:
10000:
9995:
9991:
9986:
9982:
9976:
9972:
9946:Brownian motion
9943:
9939:
9934:
9929:
9865:Ergodic process
9834:
9813:
9810:
9809:
9787:
9763:
9760:
9759:
9743:
9740:
9739:
9709:
9706:
9705:
9685:
9682:
9681:
9656:
9653:
9652:
9632:
9629:
9628:
9603:
9600:
9599:
9579:
9576:
9575:
9555:
9552:
9551:
9535:
9532:
9531:
9523:
9502:
9477:
9450:Maurice Fréchet
9416:, and proved a
9397:
9345:
9343:Poisson process
9310:Albert Einstein
9284:Louis Bachelier
9276:Albert Einstein
9272:Louis Bachelier
9268:Thorvald Thiele
9249:Thorvald Thiele
9241:
9211:
9208:
9207:
9183:
9175:Ars Conjectandi
9171:Jacob Bernoulli
9166:
9157:
9144:in 2008 and to
9142:Wendelin Werner
9086:Shizuo Kakutani
9073:and stochastic
9058:
9010:Maurice Fréchet
8991:
8983:Nikolai Smirnov
8928:
8916:Albert Einstein
8899:Rudolf Clausius
8891:
8863:Ars Conjectandi
8859:Jakob Bernoulli
8857:After Cardano,
8831:
8826:
8805:
8802:
8801:
8770:
8767:
8766:
8763:
8757:
8736:
8733:
8732:
8729:
8723:
8680:
8677:
8676:
8666:
8639:
8635:
8634:
8630:
8619:
8615:
8614:
8610:
8593:
8589:
8588:
8584:
8573:
8569:
8568:
8564:
8562:
8559:
8558:
8534:
8531:
8530:
8513:
8509:
8494:
8490:
8482:
8479:
8478:
8462:
8459:
8458:
8442:
8439:
8438:
8430:
8424:
8376:
8372:
8363:
8359:
8350:
8346:
8337:
8333:
8324:
8320:
8311:
8307:
8305:
8302:
8301:
8278:
8274:
8265:
8261:
8252:
8248:
8246:
8243:
8242:
8227:
8221:
8200:
8197:
8196:
8144:
8138:
8133:
8120:
8083:
8080:
8079:
8045:
8042:
8041:
8025:
8022:
8021:
8005:
8002:
8001:
7965:
7962:
7961:
7933:
7930:
7929:
7922:Skorokhod space
7918:
7916:Skorokhod space
7912:
7910:Skorokhod space
7889:
7885:
7876:
7872:
7853:
7849:
7840:
7836:
7823:
7818:
7817:
7813:
7801:
7791:
7787:
7783:
7770:
7766:
7762:
7760:
7757:
7756:
7724:
7720:
7713:
7711:
7702:
7698:
7671:
7667:
7658:
7654:
7641:
7636:
7635:
7631:
7629:
7626:
7625:
7600:
7596:
7592:
7590:
7587:
7586:
7565:
7561:
7557:
7555:
7552:
7551:
7548:
7527:
7524:
7523:
7507:
7504:
7503:
7500:
7477:
7473:
7464:
7460:
7441:
7437:
7428:
7424:
7411:
7406:
7405:
7401:
7389:
7379:
7375:
7371:
7358:
7354:
7350:
7348:
7345:
7344:
7311:
7307:
7298:
7294:
7282:
7278:
7271:
7267:
7253:
7249:
7240:
7236:
7224:
7220:
7213:
7209:
7208:
7204:
7186:
7182:
7173:
7169:
7156:
7151:
7150:
7146:
7144:
7141:
7140:
7115:
7111:
7107:
7105:
7102:
7101:
7080:
7076:
7072:
7070:
7067:
7066:
7063:
7033:
7029:
7005:
7001:
6994:
6990:
6988:
6985:
6984:
6959:
6955:
6931:
6927:
6920:
6916:
6914:
6911:
6910:
6887:
6883:
6868:
6864:
6862:
6859:
6858:
6842:
6834:
6831:
6830:
6810:
6807:
6806:
6780:
6779:
6768:
6765:
6764:
6748:
6745:
6744:
6728:
6725:
6724:
6721:
6693:
6690:
6689:
6672:
6668:
6666:
6663:
6662:
6634:
6622:
6618:
6613:
6610:
6609:
6575:
6563:
6559:
6554:
6551:
6550:
6505:
6502:
6501:
6479:
6476:
6475:
6449:
6445:
6437:
6434:
6433:
6410:
6406:
6404:
6401:
6400:
6378:
6375:
6374:
6358:
6355:
6354:
6328:
6327:
6316:
6313:
6312:
6296:
6293:
6292:
6286:separable space
6278:
6257:
6254:
6253:
6237:
6234:
6233:
6213:
6210:
6209:
6193:
6190:
6189:
6186:
6150:
6144:
6140:
6131:
6127:
6119:
6116:
6115:
6093:
6090:
6089:
6073:
6070:
6069:
6043:
6042:
6031:
6028:
6027:
6011:
6008:
6007:
5991:
5988:
5987:
5984:
5959:
5956:
5955:
5928:
5900:
5896:
5887:
5883:
5875:
5872:
5871:
5847:
5844:
5843:
5827:
5824:
5823:
5807:
5804:
5803:
5777:
5776:
5765:
5762:
5761:
5745:
5742:
5741:
5725:
5722:
5721:
5705:
5702:
5701:
5694:
5673:
5670:
5669:
5652:
5646:
5645:
5644:
5642:
5639:
5638:
5621:
5617:
5615:
5612:
5611:
5595:
5592:
5591:
5574:
5568:
5567:
5566:
5564:
5561:
5560:
5544:
5541:
5540:
5518:
5515:
5514:
5497:
5493:
5491:
5488:
5487:
5471:
5468:
5467:
5451:
5448:
5447:
5419:
5416:
5415:
5393:
5390:
5389:
5372:
5371:
5362:
5356:
5355:
5354:
5345:
5339:
5338:
5337:
5335:
5332:
5331:
5305:
5304:
5293:
5290:
5289:
5266:
5262:
5256:
5250:
5249:
5248:
5243:
5240:
5239:
5228:
5185:
5182:
5181:
5165:
5162:
5161:
5138:
5134:
5132:
5129:
5128:
5111:
5107:
5105:
5102:
5101:
5079:
5076:
5075:
5059:
5056:
5055:
5039:
5036:
5035:
5014:
5011:
5010:
5000:
4979:
4975:
4974:
4970:
4956:
4952:
4951:
4947:
4945:
4942:
4941:
4923:
4920:
4919:
4902:
4898:
4883:
4879:
4877:
4874:
4873:
4857:
4854:
4853:
4836:
4832:
4830:
4827:
4826:
4804:
4801:
4800:
4784:
4781:
4780:
4774:
4768:
4760:
4741:
4740:
4734:
4733:
4721:
4720:
4703:
4699:
4698:
4694:
4668:
4664:
4663:
4659:
4653:
4652:
4634:
4633:
4627:
4626:
4603:
4599:
4584:
4580:
4579:
4575:
4573:
4570:
4569:
4551:
4548:
4547:
4512:
4508:
4506:
4503:
4502:
4500:Cartesian power
4483:
4480:
4479:
4463:
4460:
4459:
4440:
4437:
4436:
4420:
4417:
4416:
4392:
4388:
4387:
4362:
4358:
4357:
4346:
4343:
4342:
4323:
4319:
4301:
4297:
4296:
4292:
4290:
4287:
4286:
4283:
4261:
4257:
4247:
4243:
4242:
4217:
4213:
4212:
4186:
4182:
4167:
4163:
4162:
4158:
4156:
4153:
4152:
4149:is defined as:
4127:
4123:
4108:
4104:
4102:
4099:
4098:
4078:
4075:
4074:
4058:
4055:
4054:
4051:
4045:
4029:probability law
4025:
3950:
3947:
3946:
3928:
3925:
3924:
3921:
3851:
3847:
3845:
3842:
3841:
3823:
3820:
3819:
3802:
3798:
3796:
3793:
3792:
3776:
3773:
3772:
3747:
3744:
3743:
3727:
3724:
3723:
3706:
3702:
3700:
3697:
3696:
3680:
3677:
3676:
3659:
3655:
3653:
3650:
3649:
3629:
3625:
3623:
3620:
3619:
3603:
3600:
3599:
3583:
3580:
3579:
3576:
3554:
3550:
3536:
3533:
3532:
3510:
3507:
3506:
3476:
3475:
3464:
3461:
3460:
3443:
3439:
3425:
3422:
3421:
3418:
3413:
3388:
3385:
3384:
3368:
3365:
3364:
3348:
3345:
3344:
3325:
3321:
3320:
3316:
3305:
3301:
3300:
3296:
3294:
3291:
3290:
3273:
3269:
3260:
3256:
3254:
3251:
3250:
3215:
3211:
3209:
3206:
3205:
3170:
3166:
3164:
3161:
3160:
3126:
3123:
3122:
3106:
3103:
3102:
3059:
3056:
3055:
3047:
3014:
3011:
3010:
2994:
2991:
2990:
2968:
2965:
2964:
2915:
2912:
2911:
2891:
2888:
2887:
2880:
2835:
2832:
2831:
2804:
2801:
2800:
2751:
2748:
2747:
2740:sample function
2736:
2734:Sample function
2711:
2708:
2707:
2679:
2676:
2675:
2669:
2642:
2639:
2638:
2622:
2619:
2618:
2601:
2596:
2595:
2593:
2590:
2589:
2569:
2566:
2565:
2549:
2546:
2545:
2542:natural numbers
2513:
2510:
2509:
2506:
2485:
2482:
2481:
2480:into the space
2465:
2462:
2461:
2440:
2436:
2434:
2431:
2430:
2413:
2409:
2407:
2404:
2403:
2378:
2375:
2374:
2352:
2349:
2348:
2299:
2296:
2295:
2279:
2276:
2275:
2250:
2247:
2246:
2224:
2221:
2220:
2217:
2172:
2169:
2168:
2150:
2147:
2146:
2118:
2115:
2114:
2088:
2087:
2076:
2073:
2072:
2053:
2050:
2049:
2033:
2030:
2029:
2009:
2006:
2005:
1989:
1986:
1985:
1965:
1962:
1961:
1941:
1938:
1937:
1920:
1919:
1917:
1914:
1913:
1893:
1890:
1889:
1863:
1862:
1851:
1848:
1847:
1841:
1836:
1815:
1812:
1811:
1792:
1789:
1788:
1776:
1774:Poisson process
1770:
1768:Poisson process
1731:
1728:
1727:
1711:
1708:
1707:
1686:
1683:
1682:
1673:
1669:
1639:
1633:
1606:
1603:
1602:
1580:
1577:
1576:
1560:
1557:
1556:
1538:
1532:
1524:Bernoulli trial
1507:
1504:
1503:
1481:
1478:
1477:
1461:
1458:
1457:
1446:
1440:
1435:
1398:
1394:
1389:
1386:
1385:
1351:
1348:
1347:
1331:
1328:
1327:
1310:
1306:
1304:
1301:
1300:
1275:
1272:
1271:
1270:. For example,
1242:
1239:
1238:
1222:
1219:
1218:
1187:
1184:
1183:
1164:
1160:
1155:
1152:
1151:
1128:
1124:
1118:
1114:
1109:
1106:
1105:
1082:
1078:
1061:
1058:
1057:
1054:
1028:
1023:
1022:
1020:
1017:
1016:
1000:
997:
996:
980:random function
964:
912:Jakob Bernoulli
908:Ars Conjectandi
888:
863:
860:
859:
839:
836:
835:
819:
816:
815:
793:random sequence
759:
757:Classifications
746:sample function
731:sample function
706:
703:
702:
691:natural numbers
674:
591:Euclidean space
587:Cartesian plane
555:random function
543:Poisson process
535:Louis Bachelier
459:population, an
383:
379:
364:Brownian motion
348:
196:Random variable
147:Bernoulli trial
28:
23:
22:
15:
12:
11:
5:
22383:
22373:
22372:
22367:
22362:
22345:
22344:
22342:
22341:
22328:
22320:
22317:
22316:
22314:
22313:
22308:
22303:
22298:
22297:
22296:
22285:
22283:
22279:
22278:
22276:
22275:
22269:
22267:
22263:
22262:
22260:
22259:
22252:
22247:
22242:
22237:
22231:
22229:
22225:
22224:
22222:
22221:
22216:
22211:
22206:
22201:
22196:
22191:
22185:
22183:
22177:
22176:
22173:
22172:
22170:
22169:
22164:
22159:
22154:
22149:
22144:
22139:
22134:
22129:
22123:
22121:
22112:
22111:
22110:
22109:
22104:
22094:
22093:
22092:
22087:
22077:
22072:
22071:
22070:
22060:
22059:
22058:
22053:
22048:
22043:
22038:
22033:
22028:
22018:
22017:
22016:
22011:
22000:
21998:
21990:
21989:
21987:
21986:
21981:
21976:
21967:
21956:
21954:
21948:
21947:
21945:
21944:
21939:
21938:
21937:
21932:
21927:
21922:
21912:
21911:
21910:
21905:
21900:
21895:
21885:
21884:
21883:
21878:
21873:
21868:
21858:
21857:
21856:
21846:
21845:
21844:
21839:
21829:
21828:
21827:
21825:Control theory
21822:
21812:
21811:
21810:
21805:
21800:
21790:
21789:
21788:
21783:
21778:
21768:
21767:
21766:
21756:
21750:
21748:
21742:
21741:
21739:
21738:
21733:
21728:
21723:
21718:
21712:
21710:
21704:
21703:
21700:
21699:
21697:
21696:
21691:
21686:
21681:
21676:
21671:
21666:
21661:
21656:
21650:
21648:
21639:
21638:
21633:
21628:
21623:
21618:
21617:
21616:
21606:
21601:
21596:
21591:
21586:
21585:
21584:
21579:
21568:
21566:
21560:
21559:
21552:
21551:
21544:
21537:
21529:
21520:
21519:
21517:
21516:
21511:
21509:List of topics
21505:
21502:
21501:
21499:
21498:
21493:
21488:
21483:
21478:
21473:
21468:
21466:Renewal theory
21463:
21458:
21453:
21448:
21443:
21438:
21433:
21431:Ergodic theory
21428:
21423:
21421:Control theory
21418:
21412:
21410:
21406:
21405:
21403:
21402:
21401:
21400:
21395:
21385:
21380:
21375:
21370:
21365:
21364:
21363:
21353:
21351:Snell envelope
21348:
21343:
21338:
21333:
21328:
21323:
21318:
21313:
21308:
21303:
21298:
21293:
21288:
21283:
21278:
21273:
21268:
21263:
21258:
21253:
21248:
21243:
21238:
21233:
21228:
21223:
21217:
21215:
21211:
21210:
21208:
21207:
21202:
21197:
21192:
21187:
21181:
21179:
21173:
21172:
21170:
21169:
21150:BorelâCantelli
21139:
21134:
21129:
21124:
21119:
21114:
21109:
21104:
21099:
21094:
21088:
21086:
21085:Limit theorems
21082:
21081:
21079:
21078:
21073:
21068:
21063:
21058:
21053:
21048:
21043:
21038:
21033:
21028:
21023:
21018:
21013:
21008:
21002:
21000:
20996:
20995:
20993:
20992:
20987:
20982:
20977:
20972:
20967:
20961:
20959:
20953:
20952:
20950:
20949:
20944:
20939:
20934:
20928:
20926:
20920:
20919:
20917:
20916:
20911:
20906:
20901:
20896:
20891:
20886:
20881:
20876:
20871:
20866:
20861:
20856:
20851:
20846:
20841:
20836:
20831:
20826:
20820:
20818:
20812:
20811:
20809:
20808:
20803:
20798:
20793:
20788:
20783:
20777:
20775:
20769:
20768:
20766:
20765:
20760:
20755:
20754:
20753:
20748:
20738:
20733:
20728:
20723:
20722:
20721:
20716:
20706:
20704:Hopfield model
20701:
20696:
20691:
20685:
20683:
20679:
20678:
20676:
20675:
20670:
20665:
20660:
20655:
20650:
20649:
20648:
20643:
20638:
20633:
20623:
20621:Markov process
20618:
20613:
20608:
20602:
20600:
20596:
20595:
20593:
20592:
20590:Wiener sausage
20587:
20585:Wiener process
20582:
20577:
20572:
20567:
20565:Stable process
20562:
20557:
20555:Semimartingale
20552:
20547:
20546:
20545:
20540:
20530:
20525:
20520:
20515:
20510:
20505:
20500:
20498:Jump diffusion
20495:
20490:
20485:
20480:
20475:
20473:Hawkes process
20470:
20465:
20460:
20455:
20453:Feller process
20450:
20445:
20440:
20435:
20430:
20425:
20420:
20418:Cauchy process
20415:
20414:
20413:
20408:
20403:
20398:
20393:
20383:
20382:
20381:
20371:
20369:Bessel process
20366:
20360:
20358:
20352:
20351:
20349:
20348:
20347:
20346:
20341:
20336:
20331:
20321:
20316:
20311:
20306:
20301:
20296:
20291:
20285:
20283:
20277:
20276:
20269:
20268:
20261:
20254:
20246:
20240:
20239:
20225:
20224:External links
20222:
20221:
20220:
20204:
20198:
20184:Emanuel Parzen
20180:
20174:
20159:
20153:
20135:
20129:
20114:
20104:Joseph L. Doob
20100:
20094:
20079:
20073:
20058:
20052:
20035:
20032:
20031:
20030:
20017:
20003:
19982:
19956:(2): 253â268.
19945:
19925:(4): 509â546.
19910:
19895:
19892:
19889:
19888:
19850:
19848:
19841:
19835:
19832:
19829:
19828:
19821:
19801:
19794:
19774:
19767:
19747:
19740:
19717:
19710:
19690:
19683:
19663:
19656:
19636:
19629:
19609:
19602:
19582:
19575:
19555:
19548:
19528:
19521:
19496:
19489:
19469:
19433:
19426:
19406:
19379:(5): 398â396.
19363:
19356:
19336:
19329:
19309:
19302:
19276:
19249:(3): 291â292.
19233:
19203:
19196:
19173:
19166:
19137:
19118:(sup3): 6â12.
19102:
19088:978-0444500144
19087:
19061:
19054:
19034:
18991:
18948:
18914:(3): 431â451.
18882:
18848:(3): 339â353.
18822:
18793:
18752:
18725:(3): 319â320.
18706:
18663:
18644:
18637:
18617:
18610:
18590:
18583:
18563:
18556:
18536:
18529:
18509:
18502:
18482:
18475:
18452:
18445:
18425:
18418:
18398:
18379:
18360:
18353:
18333:
18314:
18279:
18258:(3): 822â832.
18238:
18231:
18211:
18204:
18184:
18177:
18168:LĂ©vy Processes
18157:
18138:
18110:
18089:(1): 257â266.
18066:
18000:
17979:(3): 929â934.
17959:
17915:
17909:978-0471667193
17908:
17879:
17873:978-0471667193
17872:
17846:
17803:
17760:
17741:(4): 243â255.
17725:
17682:
17655:(3): 150â151.
17639:
17596:
17575:(4): 509â546.
17536:
17509:(7): 586â595.
17493:
17447:
17428:(1): 145â156.
17401:
17355:
17348:
17328:
17321:
17301:
17294:
17274:
17253:(2): 180â202.
17233:
17226:
17206:
17200:978-0471667193
17199:
17167:
17160:
17140:
17097:
17090:
17070:
17063:
17043:
17036:
17016:
17009:
16989:
16982:
16962:
16955:
16937:Isham, Valerie
16919:
16912:
16892:
16885:
16865:
16858:
16838:
16831:
16811:
16804:
16784:
16777:
16757:
16750:
16730:
16706:
16699:
16690:LĂ©vy Processes
16672:
16665:
16645:
16638:
16618:
16557:
16550:
16530:
16523:
16503:
16496:
16476:
16469:
16449:
16442:
16419:
16412:
16392:
16385:
16365:
16358:
16331:
16324:
16301:
16294:
16274:
16257:
16250:
16225:
16218:
16193:
16186:
16166:
16159:
16139:
16132:
16112:
16105:
16085:
16078:
16058:
16051:
16031:
16024:
16004:
15997:
15977:
15970:
15950:
15943:
15923:
15916:
15896:
15889:
15866:
15859:
15836:
15829:
15809:
15802:
15782:
15775:
15755:
15748:
15728:
15721:
15701:
15694:
15674:
15667:
15647:
15640:
15620:
15613:
15593:
15586:
15566:
15559:
15534:
15527:
15504:
15497:
15472:
15465:
15438:
15423:
15416:
15389:
15375:
15362:
15355:
15335:
15318:
15306:
15299:
15276:
15269:
15246:
15239:
15219:
15212:
15189:
15182:
15162:
15155:
15124:
15117:
15090:
15083:
15063:
15056:
15036:
15029:
15009:
15002:
14982:
14975:
14952:
14945:
14922:
14915:
14895:
14888:
14868:
14861:
14838:
14831:
14811:
14804:
14784:
14777:
14752:
14745:
14725:
14718:
14695:
14675:
14668:
14648:
14641:
14614:
14607:
14587:
14580:
14553:
14546:
14526:
14519:
14499:
14492:
14472:
14465:
14445:
14438:
14418:
14411:
14379:
14372:
14352:
14345:
14325:
14318:
14298:
14291:
14271:
14264:
14244:
14237:
14212:
14171:
14164:
14144:
14137:
14111:
14104:
14084:
14077:
14054:
14047:
14027:
14020:
13997:
13990:
13967:
13944:
13937:
13911:
13904:
13884:
13877:
13857:
13850:
13830:
13823:
13803:
13796:
13776:
13769:
13749:
13742:
13717:
13710:
13690:
13683:
13663:
13656:
13636:
13629:
13609:
13590:
13583:
13563:
13556:
13536:
13529:
13509:
13502:
13482:
13475:
13455:
13448:
13428:
13421:
13401:
13394:
13374:
13367:
13344:
13337:
13317:
13310:
13290:
13283:
13263:
13256:
13233:
13226:
13206:
13199:
13176:
13169:
13149:
13142:
13122:
13103:
13060:
13053:
13033:
13026:
13006:
12999:
12979:
12972:
12949:
12942:
12922:
12916:978-0471667193
12915:
12884:
12877:
12857:
12850:
12830:
12823:
12803:
12796:
12776:
12769:
12749:
12742:
12722:
12715:
12692:
12685:
12662:
12655:
12632:
12625:
12605:
12598:
12574:
12567:
12547:
12532:
12525:
12505:
12498:
12478:
12471:
12448:
12441:
12421:
12398:
12391:
12368:
12361:
12332:
12325:
12298:
12291:
12271:
12237:
12194:
12167:(1): 604â615.
12151:
12112:(6): 376â379.
12085:
12078:
12058:
12051:
12031:
11994:
11987:
11959:
11952:
11929:
11922:
11893:
11886:
11861:
11854:
11831:
11824:
11801:
11794:
11774:
11767:
11744:
11737:
11706:
11699:
11676:
11638:
11631:
11604:
11597:
11574:
11567:
11547:
11540:
11513:
11494:
11487:
11467:
11460:
11440:
11433:
11413:
11406:
11386:
11379:
11356:
11349:
11329:
11322:
11302:
11295:
11275:
11268:
11248:
11241:
11221:
11214:
11194:
11187:
11167:
11160:
11140:
11133:
11098:
11091:
11064:
11057:
11006:
10999:
10968:
10961:
10941:
10934:
10910:
10883:(2): 253â268.
10860:
10853:
10833:
10767:
10752:
10735:10.1.1.114.632
10699:
10680:
10673:
10653:
10646:
10622:
10615:
10588:
10581:
10557:
10550:
10542:Now Publishers
10526:
10519:
10495:
10488:
10482:. p. 71.
10464:
10457:
10434:
10427:
10403:
10396:
10372:
10365:
10341:
10334:
10310:
10303:
10279:
10272:
10249:
10242:
10199:
10192:
10163:
10156:
10131:
10124:
10097:
10062:
10061:
10059:
10056:
10053:
10052:
10043:
10034:
10025:
10016:
10007:
9998:
9989:
9980:
9970:
9968:or Rosenblatt.
9958:Wiener process
9936:
9935:
9933:
9930:
9928:
9927:
9922:
9917:
9912:
9907:
9902:
9897:
9892:
9887:
9882:
9877:
9872:
9867:
9862:
9856:
9851:
9846:
9841:
9835:
9833:
9830:
9817:
9786:
9783:
9767:
9747:
9732:
9731:
9719:
9716:
9713:
9689:
9678:
9666:
9663:
9660:
9636:
9625:
9613:
9610:
9607:
9583:
9559:
9539:
9522:
9519:
9501:
9498:
9476:
9475:LĂ©vy processes
9473:
9465:Sydney Chapman
9461:Norbert Wiener
9437:Francis Galton
9396:
9393:
9357:Filip Lundberg
9349:Siméon Poisson
9344:
9341:
9303:Leonard Savage
9296:Jules Regnault
9292:stock exchange
9264:Norbert Wiener
9245:Wiener process
9240:
9239:Wiener process
9237:
9215:
9196:Gambler's ruin
9182:
9179:
9165:
9162:
9156:
9153:
9057:
9054:
9043:Mathematician
9006:William Feller
8990:
8987:
8979:Eugene Slutsky
8952:Henri Lebesgue
8927:
8924:
8890:
8887:
8879:Siméon Poisson
8830:
8827:
8825:
8822:
8809:
8774:
8759:Main article:
8756:
8753:
8740:
8725:Main article:
8722:
8719:
8702:
8699:
8696:
8693:
8690:
8687:
8684:
8655:
8648:
8645:
8642:
8638:
8633:
8629:
8622:
8618:
8613:
8609:
8606:
8603:
8596:
8592:
8587:
8583:
8576:
8572:
8567:
8557:
8544:
8541:
8538:
8516:
8512:
8508:
8505:
8502:
8497:
8493:
8489:
8486:
8466:
8446:
8426:Main article:
8423:
8420:
8387:
8384:
8379:
8375:
8371:
8366:
8362:
8358:
8353:
8349:
8345:
8340:
8336:
8332:
8327:
8323:
8319:
8314:
8310:
8289:
8286:
8281:
8277:
8273:
8268:
8264:
8260:
8255:
8251:
8223:Main article:
8220:
8217:
8204:
8159:gambler's ruin
8140:Main article:
8137:
8134:
8132:
8129:
8119:
8116:
8099:
8096:
8093:
8090:
8087:
8064:
8061:
8058:
8055:
8052:
8049:
8029:
8009:
7981:
7978:
7975:
7972:
7969:
7949:
7946:
7943:
7940:
7937:
7926:Skorohod space
7914:Main article:
7911:
7908:
7907:
7906:
7892:
7888:
7884:
7879:
7875:
7871:
7867:
7864:
7861:
7856:
7852:
7848:
7843:
7839:
7835:
7832:
7826:
7821:
7816:
7810:
7799:
7794:
7790:
7786:
7782:
7778:
7773:
7769:
7765:
7741:
7736:
7732:
7727:
7723:
7719:
7716:
7710:
7705:
7701:
7697:
7694:
7691:
7688:
7685:
7682:
7679:
7674:
7670:
7666:
7661:
7657:
7653:
7650:
7644:
7639:
7634:
7608:
7603:
7599:
7595:
7573:
7568:
7564:
7560:
7547:
7544:
7531:
7511:
7499:
7496:
7495:
7494:
7480:
7476:
7472:
7467:
7463:
7459:
7455:
7452:
7449:
7444:
7440:
7436:
7431:
7427:
7423:
7420:
7414:
7409:
7404:
7398:
7387:
7382:
7378:
7374:
7370:
7366:
7361:
7357:
7353:
7328:
7323:
7319:
7314:
7310:
7306:
7301:
7297:
7293:
7290:
7285:
7281:
7277:
7274:
7270:
7265:
7261:
7256:
7252:
7248:
7243:
7239:
7235:
7232:
7227:
7223:
7219:
7216:
7212:
7207:
7203:
7200:
7197:
7194:
7189:
7185:
7181:
7176:
7172:
7168:
7165:
7159:
7154:
7149:
7123:
7118:
7114:
7110:
7088:
7083:
7079:
7075:
7062:
7059:
7045:
7041:
7036:
7032:
7028:
7025:
7022:
7019:
7016:
7013:
7008:
7004:
7000:
6997:
6993:
6971:
6967:
6962:
6958:
6954:
6951:
6948:
6945:
6942:
6939:
6934:
6930:
6926:
6923:
6919:
6898:
6895:
6890:
6886:
6882:
6879:
6876:
6871:
6867:
6845:
6841:
6838:
6814:
6794:
6791:
6788:
6783:
6778:
6775:
6772:
6752:
6732:
6720:
6717:
6697:
6675:
6671:
6650:
6647:
6644:
6641:
6633:
6630:
6625:
6621:
6617:
6597:
6594:
6591:
6588:
6585:
6582:
6574:
6571:
6566:
6562:
6558:
6537:
6534:
6531:
6528:
6525:
6522:
6519:
6516:
6512:
6509:
6489:
6486:
6483:
6463:
6460:
6457:
6452:
6448:
6444:
6441:
6421:
6418:
6413:
6409:
6388:
6385:
6382:
6362:
6342:
6339:
6336:
6331:
6326:
6323:
6320:
6300:
6277:
6274:
6261:
6241:
6217:
6197:
6188:holds. If two
6175:
6172:
6169:
6166:
6163:
6160:
6157:
6147:
6143:
6139:
6134:
6130:
6126:
6123:
6114:
6097:
6088:and set space
6077:
6057:
6054:
6051:
6046:
6041:
6038:
6035:
6015:
5995:
5983:
5980:
5963:
5917:
5914:
5911:
5908:
5903:
5899:
5895:
5890:
5886:
5882:
5879:
5870:
5868:the following
5857:
5854:
5851:
5831:
5811:
5791:
5788:
5785:
5780:
5775:
5772:
5769:
5749:
5740:, state space
5729:
5709:
5693:
5690:
5677:
5655:
5649:
5624:
5620:
5599:
5577:
5571:
5548:
5528:
5525:
5522:
5500:
5496:
5475:
5455:
5435:
5432:
5429:
5426:
5423:
5403:
5400:
5397:
5375:
5370:
5365:
5359:
5353:
5348:
5342:
5319:
5316:
5313:
5308:
5303:
5300:
5297:
5275:
5272:
5269:
5265:
5259:
5253:
5247:
5227:
5224:
5195:
5192:
5189:
5169:
5147:
5144:
5141:
5137:
5114:
5110:
5089:
5086:
5083:
5063:
5043:
5018:
4989:
4982:
4978:
4973:
4969:
4966:
4959:
4955:
4950:
4940:
4927:
4905:
4901:
4897:
4894:
4891:
4886:
4882:
4861:
4839:
4835:
4814:
4811:
4808:
4788:
4770:Main article:
4767:
4764:
4749:
4744:
4737:
4732:
4729:
4724:
4719:
4716:
4713:
4706:
4702:
4697:
4693:
4690:
4687:
4684:
4681:
4678:
4671:
4667:
4662:
4656:
4651:
4648:
4645:
4642:
4637:
4630:
4625:
4622:
4619:
4616:
4613:
4606:
4602:
4598:
4595:
4592:
4587:
4583:
4578:
4568:
4555:
4535:
4532:
4529:
4526:
4523:
4520:
4515:
4511:
4487:
4467:
4444:
4424:
4404:
4401:
4395:
4391:
4386:
4383:
4380:
4377:
4374:
4371:
4365:
4361:
4356:
4353:
4350:
4326:
4322:
4318:
4315:
4312:
4309:
4304:
4300:
4295:
4272:
4267:
4264:
4260:
4256:
4250:
4246:
4241:
4238:
4235:
4232:
4229:
4226:
4220:
4216:
4211:
4208:
4205:
4202:
4199:
4196:
4189:
4185:
4181:
4178:
4175:
4170:
4166:
4161:
4151:
4138:
4135:
4130:
4126:
4122:
4119:
4116:
4111:
4107:
4082:
4062:
4047:Main article:
4044:
4041:
4014:
4011:
4008:
4005:
4002:
3999:
3996:
3993:
3990:
3987:
3984:
3981:
3978:
3975:
3972:
3969:
3966:
3963:
3960:
3957:
3954:
3945:
3932:
3910:
3907:
3904:
3901:
3898:
3895:
3892:
3889:
3886:
3883:
3880:
3877:
3874:
3871:
3868:
3865:
3862:
3857:
3854:
3850:
3840:
3827:
3805:
3801:
3780:
3757:
3754:
3751:
3731:
3709:
3705:
3684:
3662:
3658:
3635:
3632:
3628:
3607:
3587:
3565:
3560:
3557:
3553:
3549:
3546:
3543:
3540:
3531:
3514:
3490:
3487:
3484:
3479:
3474:
3471:
3468:
3446:
3442:
3438:
3435:
3432:
3429:
3417:
3414:
3412:
3409:
3392:
3372:
3352:
3328:
3324:
3319:
3315:
3308:
3304:
3299:
3276:
3272:
3268:
3263:
3259:
3238:
3235:
3232:
3229:
3226:
3223:
3218:
3214:
3193:
3190:
3187:
3184:
3181:
3178:
3173:
3169:
3148:
3145:
3142:
3139:
3136:
3133:
3130:
3121:and index set
3110:
3090:
3087:
3084:
3081:
3078:
3075:
3072:
3069:
3066:
3063:
3046:
3043:
3018:
2998:
2978:
2975:
2972:
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2928:
2925:
2922:
2919:
2895:
2869:
2866:
2863:
2860:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2830:
2814:
2811:
2808:
2788:
2785:
2782:
2779:
2776:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2735:
2732:
2715:
2683:
2668:
2665:
2652:
2649:
2646:
2626:
2604:
2599:
2573:
2553:
2540:, such as the
2528:is called the
2517:
2505:
2502:
2489:
2469:
2443:
2439:
2416:
2412:
2388:
2385:
2382:
2362:
2359:
2356:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2306:
2303:
2283:
2263:
2260:
2257:
2254:
2234:
2231:
2228:
2206:
2203:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2179:
2176:
2167:
2154:
2134:
2131:
2128:
2125:
2122:
2102:
2099:
2096:
2091:
2086:
2083:
2080:
2057:
2037:
2013:
1993:
1969:
1945:
1923:
1897:
1877:
1874:
1871:
1866:
1861:
1858:
1855:
1840:
1837:
1835:
1832:
1819:
1796:
1772:Main article:
1769:
1766:
1735:
1715:
1690:
1655:Norbert Wiener
1637:Wiener process
1635:Main article:
1632:
1631:Wiener process
1629:
1616:
1613:
1610:
1590:
1587:
1584:
1564:
1534:Main article:
1531:
1528:
1511:
1491:
1488:
1485:
1465:
1442:Main article:
1439:
1436:
1434:
1431:
1418:
1415:
1412:
1409:
1406:
1401:
1397:
1393:
1373:
1370:
1367:
1364:
1361:
1358:
1355:
1335:
1313:
1309:
1288:
1285:
1282:
1279:
1255:
1252:
1249:
1246:
1226:
1206:
1203:
1200:
1197:
1194:
1191:
1172:
1167:
1163:
1159:
1137:
1134:
1131:
1127:
1121:
1117:
1113:
1091:
1088:
1085:
1081:
1077:
1074:
1071:
1068:
1065:
1053:
1050:
1031:
1026:
1004:
989:random process
968:random process
963:
960:
948:random process
887:
884:
880:vector process
867:
843:
823:
758:
755:
710:
689:, such as the
673:
670:
658:measure theory
638:linear algebra
614:LĂ©vy processes
567:random process
559:function space
531:Wiener process
504:control theory
421:random process
350:
349:
347:
346:
339:
332:
324:
321:
320:
319:
318:
313:
305:
304:
303:
302:
297:
295:Bayes' theorem
292:
287:
282:
277:
269:
268:
267:
266:
261:
256:
251:
243:
242:
241:
240:
239:
238:
233:
228:
226:Observed value
223:
218:
213:
211:Expected value
208:
203:
193:
188:
187:
186:
181:
176:
171:
166:
161:
151:
150:
149:
139:
138:
137:
132:
127:
122:
117:
107:
102:
94:
93:
92:
91:
86:
81:
80:
79:
69:
68:
67:
54:
53:
45:
44:
38:
37:
26:
9:
6:
4:
3:
2:
22382:
22371:
22368:
22366:
22363:
22361:
22358:
22357:
22355:
22340:
22336:
22332:
22329:
22327:
22326:
22322:
22321:
22318:
22312:
22309:
22307:
22304:
22302:
22299:
22295:
22292:
22291:
22290:
22287:
22286:
22284:
22282:Organizations
22280:
22274:
22271:
22270:
22268:
22264:
22257:
22253:
22251:
22248:
22246:
22243:
22241:
22238:
22236:
22233:
22232:
22230:
22226:
22220:
22217:
22215:
22212:
22210:
22207:
22205:
22202:
22200:
22197:
22195:
22192:
22190:
22187:
22186:
22184:
22182:
22178:
22168:
22165:
22163:
22160:
22158:
22155:
22153:
22150:
22148:
22145:
22143:
22142:Quantum group
22140:
22138:
22135:
22133:
22130:
22128:
22125:
22124:
22122:
22120:
22116:
22108:
22105:
22103:
22100:
22099:
22098:
22097:Supersymmetry
22095:
22091:
22088:
22086:
22083:
22082:
22081:
22080:String theory
22078:
22076:
22073:
22069:
22066:
22065:
22064:
22061:
22057:
22054:
22052:
22049:
22047:
22044:
22042:
22039:
22037:
22034:
22032:
22029:
22027:
22024:
22023:
22022:
22019:
22015:
22012:
22010:
22007:
22006:
22005:
22002:
22001:
21999:
21997:
21991:
21985:
21982:
21980:
21979:Path integral
21977:
21975:
21971:
21968:
21965:
21961:
21960:Distributions
21958:
21957:
21955:
21953:
21949:
21943:
21940:
21936:
21933:
21931:
21928:
21926:
21923:
21921:
21918:
21917:
21916:
21913:
21909:
21906:
21904:
21901:
21899:
21896:
21894:
21891:
21890:
21889:
21886:
21882:
21879:
21877:
21874:
21872:
21869:
21867:
21864:
21863:
21862:
21859:
21855:
21852:
21851:
21850:
21847:
21843:
21840:
21838:
21835:
21834:
21833:
21830:
21826:
21823:
21821:
21818:
21817:
21816:
21813:
21809:
21806:
21804:
21801:
21799:
21796:
21795:
21794:
21791:
21787:
21784:
21782:
21779:
21777:
21774:
21773:
21772:
21769:
21765:
21762:
21761:
21760:
21757:
21755:
21752:
21751:
21749:
21747:
21743:
21737:
21734:
21732:
21729:
21727:
21726:Combinatorics
21724:
21722:
21719:
21717:
21714:
21713:
21711:
21709:
21705:
21695:
21692:
21690:
21687:
21685:
21682:
21680:
21677:
21675:
21672:
21670:
21667:
21665:
21662:
21660:
21657:
21655:
21652:
21651:
21649:
21647:
21643:
21637:
21634:
21632:
21629:
21627:
21624:
21622:
21619:
21615:
21612:
21611:
21610:
21607:
21605:
21602:
21600:
21599:Coding theory
21597:
21595:
21592:
21590:
21587:
21583:
21580:
21578:
21575:
21574:
21573:
21570:
21569:
21567:
21565:
21564:Computational
21561:
21557:
21550:
21545:
21543:
21538:
21536:
21531:
21530:
21527:
21515:
21512:
21510:
21507:
21506:
21503:
21497:
21494:
21492:
21489:
21487:
21484:
21482:
21479:
21477:
21474:
21472:
21469:
21467:
21464:
21462:
21459:
21457:
21454:
21452:
21449:
21447:
21444:
21442:
21439:
21437:
21434:
21432:
21429:
21427:
21424:
21422:
21419:
21417:
21414:
21413:
21411:
21407:
21399:
21396:
21394:
21391:
21390:
21389:
21386:
21384:
21381:
21379:
21376:
21374:
21371:
21369:
21368:Stopping time
21366:
21362:
21359:
21358:
21357:
21354:
21352:
21349:
21347:
21344:
21342:
21339:
21337:
21334:
21332:
21329:
21327:
21324:
21322:
21319:
21317:
21314:
21312:
21309:
21307:
21304:
21302:
21299:
21297:
21294:
21292:
21289:
21287:
21284:
21282:
21279:
21277:
21274:
21272:
21269:
21267:
21264:
21262:
21259:
21257:
21254:
21252:
21249:
21247:
21244:
21242:
21239:
21237:
21234:
21232:
21229:
21227:
21224:
21222:
21219:
21218:
21216:
21212:
21206:
21203:
21201:
21198:
21196:
21193:
21191:
21188:
21186:
21183:
21182:
21180:
21178:
21174:
21167:
21163:
21159:
21158:HewittâSavage
21155:
21151:
21147:
21143:
21142:Zeroâone laws
21140:
21138:
21135:
21133:
21130:
21128:
21125:
21123:
21120:
21118:
21115:
21113:
21110:
21108:
21105:
21103:
21100:
21098:
21095:
21093:
21090:
21089:
21087:
21083:
21077:
21074:
21072:
21069:
21067:
21064:
21062:
21059:
21057:
21054:
21052:
21049:
21047:
21044:
21042:
21039:
21037:
21034:
21032:
21029:
21027:
21024:
21022:
21019:
21017:
21014:
21012:
21009:
21007:
21004:
21003:
21001:
20997:
20991:
20988:
20986:
20983:
20981:
20978:
20976:
20973:
20971:
20968:
20966:
20963:
20962:
20960:
20958:
20954:
20948:
20945:
20943:
20940:
20938:
20935:
20933:
20930:
20929:
20927:
20925:
20921:
20915:
20912:
20910:
20907:
20905:
20902:
20900:
20897:
20895:
20892:
20890:
20887:
20885:
20882:
20880:
20877:
20875:
20872:
20870:
20867:
20865:
20862:
20860:
20857:
20855:
20852:
20850:
20847:
20845:
20842:
20840:
20839:BlackâScholes
20837:
20835:
20832:
20830:
20827:
20825:
20822:
20821:
20819:
20817:
20813:
20807:
20804:
20802:
20799:
20797:
20794:
20792:
20789:
20787:
20784:
20782:
20779:
20778:
20776:
20774:
20770:
20764:
20761:
20759:
20756:
20752:
20749:
20747:
20744:
20743:
20742:
20741:Point process
20739:
20737:
20734:
20732:
20729:
20727:
20724:
20720:
20717:
20715:
20712:
20711:
20710:
20707:
20705:
20702:
20700:
20699:Gibbs measure
20697:
20695:
20692:
20690:
20687:
20686:
20684:
20680:
20674:
20671:
20669:
20666:
20664:
20661:
20659:
20656:
20654:
20651:
20647:
20644:
20642:
20639:
20637:
20634:
20632:
20629:
20628:
20627:
20624:
20622:
20619:
20617:
20614:
20612:
20609:
20607:
20604:
20603:
20601:
20597:
20591:
20588:
20586:
20583:
20581:
20578:
20576:
20573:
20571:
20568:
20566:
20563:
20561:
20558:
20556:
20553:
20551:
20548:
20544:
20541:
20539:
20536:
20535:
20534:
20531:
20529:
20526:
20524:
20521:
20519:
20516:
20514:
20511:
20509:
20506:
20504:
20501:
20499:
20496:
20494:
20491:
20489:
20488:ItĂŽ diffusion
20486:
20484:
20481:
20479:
20476:
20474:
20471:
20469:
20466:
20464:
20463:Gamma process
20461:
20459:
20456:
20454:
20451:
20449:
20446:
20444:
20441:
20439:
20436:
20434:
20431:
20429:
20426:
20424:
20421:
20419:
20416:
20412:
20409:
20407:
20404:
20402:
20399:
20397:
20394:
20392:
20389:
20388:
20387:
20384:
20380:
20377:
20376:
20375:
20372:
20370:
20367:
20365:
20362:
20361:
20359:
20357:
20353:
20345:
20342:
20340:
20337:
20335:
20334:Self-avoiding
20332:
20330:
20327:
20326:
20325:
20322:
20320:
20319:Moran process
20317:
20315:
20312:
20310:
20307:
20305:
20302:
20300:
20297:
20295:
20292:
20290:
20287:
20286:
20284:
20282:
20281:Discrete time
20278:
20274:
20267:
20262:
20260:
20255:
20253:
20248:
20247:
20244:
20237:
20232:
20228:
20227:
20216:
20215:
20209:
20205:
20201:
20195:
20191:
20190:
20185:
20181:
20177:
20171:
20167:
20166:
20160:
20156:
20150:
20146:
20145:
20140:
20136:
20132:
20126:
20122:
20121:
20115:
20111:
20110:
20105:
20101:
20097:
20091:
20087:
20086:
20080:
20076:
20070:
20066:
20065:
20059:
20055:
20049:
20045:
20044:
20038:
20037:
20027:
20023:
20018:
20014:
20010:
20006:
20000:
19996:
19992:
19988:
19983:
19979:
19975:
19971:
19967:
19963:
19959:
19955:
19951:
19946:
19942:
19938:
19933:
19928:
19924:
19920:
19916:
19911:
19907:
19903:
19898:
19897:
19885:
19882:
19874:
19864:
19863:editing guide
19858:
19854:
19849:
19840:
19839:
19824:
19818:
19814:
19813:
19805:
19797:
19791:
19787:
19786:
19778:
19770:
19764:
19760:
19759:
19751:
19743:
19737:
19733:
19732:
19724:
19722:
19713:
19707:
19703:
19702:
19694:
19686:
19680:
19676:
19675:
19667:
19659:
19653:
19649:
19648:
19640:
19632:
19626:
19622:
19621:
19613:
19605:
19599:
19595:
19594:
19586:
19578:
19572:
19568:
19567:
19559:
19551:
19545:
19541:
19540:
19532:
19524:
19518:
19514:
19513:
19505:
19503:
19501:
19492:
19486:
19482:
19481:
19473:
19465:
19461:
19457:
19453:
19449:
19445:
19437:
19429:
19423:
19419:
19418:
19410:
19402:
19398:
19394:
19390:
19386:
19382:
19378:
19374:
19367:
19359:
19353:
19349:
19348:
19340:
19332:
19326:
19322:
19321:
19313:
19305:
19299:
19295:
19291:
19287:
19280:
19272:
19268:
19264:
19260:
19256:
19252:
19248:
19244:
19237:
19229:
19225:
19221:
19217:
19210:
19208:
19199:
19193:
19189:
19188:
19180:
19178:
19169:
19163:
19159:
19155:
19154:
19146:
19144:
19142:
19133:
19129:
19125:
19121:
19117:
19113:
19106:
19098:
19094:
19090:
19084:
19080:
19076:
19072:
19065:
19057:
19051:
19047:
19046:
19038:
19030:
19026:
19022:
19018:
19014:
19010:
19006:
19002:
18995:
18987:
18983:
18979:
18975:
18971:
18967:
18963:
18959:
18952:
18941:
18937:
18933:
18929:
18925:
18921:
18917:
18913:
18909:
18902:
18895:
18893:
18891:
18889:
18887:
18875:
18871:
18867:
18863:
18859:
18855:
18851:
18847:
18843:
18836:
18829:
18827:
18817:
18812:
18808:
18804:
18797:
18786:
18781:
18776:
18772:
18771:
18763:
18756:
18748:
18744:
18740:
18736:
18732:
18728:
18724:
18720:
18713:
18711:
18702:
18698:
18694:
18690:
18686:
18682:
18678:
18674:
18667:
18659:
18655:
18648:
18640:
18634:
18630:
18629:
18621:
18613:
18607:
18603:
18602:
18594:
18586:
18580:
18576:
18575:
18567:
18559:
18553:
18549:
18548:
18540:
18532:
18526:
18522:
18521:
18513:
18505:
18499:
18495:
18494:
18486:
18478:
18472:
18468:
18467:
18459:
18457:
18448:
18442:
18438:
18437:
18429:
18421:
18415:
18411:
18410:
18402:
18394:
18390:
18383:
18375:
18371:
18364:
18356:
18350:
18346:
18345:
18337:
18330:(2): 238â246.
18329:
18325:
18318:
18310:
18306:
18302:
18298:
18294:
18290:
18283:
18275:
18271:
18266:
18261:
18257:
18253:
18249:
18242:
18234:
18228:
18224:
18223:
18215:
18207:
18201:
18197:
18196:
18188:
18180:
18174:
18170:
18169:
18161:
18153:
18149:
18142:
18134:
18130:
18123:
18121:
18119:
18117:
18115:
18106:
18102:
18097:
18092:
18088:
18084:
18080:
18073:
18071:
18062:
18058:
18054:
18050:
18046:
18042:
18038:
18034:
18029:
18024:
18020:
18016:
18009:
18007:
18005:
17996:
17992:
17987:
17982:
17978:
17974:
17970:
17963:
17955:
17951:
17946:
17941:
17937:
17933:
17929:
17922:
17920:
17911:
17905:
17901:
17897:
17894:. p. 4.
17893:
17886:
17884:
17875:
17869:
17865:
17861:
17858:. p. 1.
17857:
17850:
17842:
17838:
17834:
17830:
17826:
17822:
17814:
17812:
17810:
17808:
17799:
17795:
17791:
17787:
17783:
17779:
17775:
17771:
17764:
17756:
17752:
17748:
17744:
17740:
17736:
17729:
17721:
17717:
17713:
17709:
17705:
17701:
17697:
17693:
17686:
17678:
17674:
17670:
17666:
17662:
17658:
17654:
17650:
17643:
17635:
17631:
17627:
17623:
17619:
17615:
17611:
17607:
17600:
17592:
17588:
17583:
17578:
17574:
17570:
17566:
17559:
17557:
17555:
17553:
17551:
17549:
17547:
17545:
17543:
17541:
17532:
17528:
17524:
17520:
17516:
17512:
17508:
17504:
17497:
17489:
17485:
17481:
17477:
17473:
17469:
17465:
17461:
17454:
17452:
17443:
17439:
17435:
17431:
17427:
17423:
17416:
17414:
17412:
17410:
17408:
17406:
17397:
17393:
17389:
17385:
17381:
17377:
17373:
17369:
17362:
17360:
17351:
17345:
17341:
17340:
17332:
17324:
17318:
17314:
17313:
17305:
17297:
17291:
17287:
17286:
17278:
17270:
17266:
17261:
17256:
17252:
17248:
17244:
17237:
17229:
17223:
17219:
17218:
17210:
17202:
17196:
17192:
17188:
17185:. p. 1.
17184:
17180:
17174:
17172:
17163:
17157:
17153:
17152:
17144:
17136:
17132:
17128:
17124:
17120:
17116:
17113:(1/2): 1â15.
17112:
17108:
17101:
17093:
17087:
17083:
17082:
17074:
17066:
17060:
17056:
17055:
17047:
17039:
17033:
17029:
17028:
17020:
17012:
17006:
17002:
17001:
16993:
16985:
16979:
16975:
16974:
16966:
16958:
16952:
16948:
16945:. CRC Press.
16944:
16943:
16938:
16934:
16928:
16926:
16924:
16915:
16909:
16905:
16904:
16896:
16888:
16882:
16878:
16877:
16869:
16861:
16855:
16851:
16850:
16842:
16834:
16828:
16824:
16823:
16815:
16807:
16801:
16797:
16796:
16788:
16780:
16774:
16770:
16769:
16761:
16753:
16747:
16743:
16742:
16734:
16726:
16722:
16715:
16713:
16711:
16702:
16696:
16692:
16691:
16683:
16681:
16679:
16677:
16668:
16662:
16658:
16657:
16649:
16641:
16635:
16631:
16630:
16622:
16614:
16610:
16606:
16602:
16598:
16594:
16590:
16586:
16581:
16576:
16572:
16568:
16561:
16553:
16547:
16543:
16542:
16534:
16526:
16520:
16516:
16515:
16507:
16499:
16493:
16489:
16488:
16480:
16472:
16466:
16462:
16461:
16453:
16445:
16439:
16435:
16434:
16426:
16424:
16415:
16409:
16405:
16404:
16396:
16388:
16382:
16378:
16377:
16369:
16361:
16355:
16351:
16350:
16345:
16338:
16336:
16327:
16321:
16317:
16316:
16308:
16306:
16297:
16291:
16287:
16286:
16278:
16270:
16269:
16261:
16253:
16247:
16243:
16242:
16234:
16232:
16230:
16221:
16215:
16211:
16210:
16202:
16200:
16198:
16189:
16183:
16179:
16178:
16170:
16162:
16156:
16152:
16151:
16143:
16135:
16129:
16125:
16124:
16116:
16108:
16102:
16099:. CRC Press.
16098:
16097:
16089:
16081:
16075:
16071:
16070:
16062:
16054:
16048:
16044:
16043:
16035:
16027:
16021:
16017:
16016:
16008:
16000:
15994:
15990:
15989:
15981:
15973:
15967:
15963:
15962:
15954:
15946:
15940:
15936:
15935:
15927:
15919:
15913:
15909:
15908:
15900:
15892:
15886:
15882:
15881:
15873:
15871:
15862:
15856:
15852:
15851:
15843:
15841:
15832:
15826:
15822:
15821:
15813:
15805:
15799:
15795:
15794:
15786:
15778:
15772:
15768:
15767:
15759:
15751:
15745:
15741:
15740:
15732:
15724:
15718:
15714:
15713:
15705:
15697:
15691:
15687:
15686:
15678:
15670:
15664:
15660:
15659:
15651:
15643:
15637:
15633:
15632:
15624:
15616:
15610:
15606:
15605:
15597:
15589:
15583:
15579:
15578:
15570:
15562:
15556:
15552:
15551:
15543:
15541:
15539:
15530:
15524:
15520:
15519:
15511:
15509:
15500:
15494:
15490:
15489:
15481:
15479:
15477:
15468:
15462:
15458:
15457:
15449:
15447:
15445:
15443:
15435:
15430:
15428:
15419:
15413:
15409:
15408:
15400:
15398:
15396:
15394:
15384:
15382:
15380:
15372:
15366:
15358:
15352:
15348:
15347:
15339:
15331:
15330:
15322:
15315:
15310:
15302:
15296:
15292:
15291:
15283:
15281:
15272:
15266:
15262:
15261:
15253:
15251:
15242:
15236:
15232:
15231:
15223:
15215:
15209:
15205:
15204:
15196:
15194:
15185:
15179:
15175:
15174:
15166:
15158:
15152:
15148:
15147:
15139:
15137:
15135:
15133:
15131:
15129:
15120:
15114:
15110:
15109:
15104:
15100:
15094:
15086:
15080:
15076:
15075:
15067:
15059:
15053:
15049:
15048:
15040:
15032:
15026:
15022:
15021:
15013:
15005:
14999:
14995:
14994:
14986:
14978:
14972:
14968:
14967:
14959:
14957:
14948:
14942:
14938:
14937:
14929:
14927:
14918:
14912:
14908:
14907:
14899:
14891:
14885:
14881:
14880:
14872:
14864:
14858:
14854:
14853:
14845:
14843:
14834:
14828:
14824:
14823:
14815:
14807:
14801:
14797:
14796:
14788:
14780:
14774:
14770:
14769:
14761:
14759:
14757:
14748:
14742:
14738:
14737:
14729:
14721:
14715:
14711:
14710:
14702:
14700:
14691:
14690:
14682:
14680:
14671:
14665:
14661:
14660:
14652:
14644:
14638:
14634:
14633:
14625:
14623:
14621:
14619:
14610:
14604:
14600:
14599:
14591:
14583:
14577:
14573:
14572:
14564:
14562:
14560:
14558:
14549:
14543:
14539:
14538:
14530:
14522:
14516:
14512:
14511:
14503:
14495:
14489:
14485:
14484:
14476:
14468:
14462:
14458:
14457:
14449:
14441:
14435:
14431:
14430:
14422:
14414:
14408:
14404:
14403:
14398:
14397:Peter K. Friz
14392:
14390:
14388:
14386:
14384:
14375:
14369:
14365:
14364:
14356:
14348:
14342:
14338:
14337:
14329:
14321:
14315:
14311:
14310:
14302:
14294:
14288:
14284:
14283:
14275:
14267:
14261:
14257:
14256:
14248:
14240:
14234:
14230:
14229:
14221:
14219:
14217:
14208:
14204:
14199:
14194:
14190:
14186:
14182:
14175:
14167:
14161:
14157:
14156:
14148:
14140:
14134:
14130:
14129:
14124:
14118:
14116:
14107:
14101:
14097:
14096:
14088:
14080:
14074:
14070:
14069:
14061:
14059:
14050:
14044:
14040:
14039:
14031:
14023:
14017:
14013:
14012:
14004:
14002:
13993:
13987:
13983:
13982:
13974:
13972:
13963:
13958:
13957:
13948:
13940:
13934:
13930:
13925:
13924:
13915:
13907:
13901:
13897:
13896:
13888:
13880:
13874:
13870:
13869:
13861:
13853:
13847:
13843:
13842:
13834:
13826:
13820:
13816:
13815:
13807:
13799:
13793:
13789:
13788:
13780:
13772:
13766:
13762:
13761:
13753:
13745:
13739:
13735:
13734:
13726:
13724:
13722:
13713:
13707:
13703:
13702:
13694:
13686:
13680:
13676:
13675:
13667:
13659:
13653:
13649:
13648:
13640:
13632:
13626:
13622:
13621:
13613:
13605:
13601:
13594:
13586:
13580:
13576:
13575:
13567:
13559:
13553:
13549:
13548:
13540:
13532:
13526:
13522:
13521:
13513:
13505:
13499:
13495:
13494:
13486:
13478:
13472:
13468:
13467:
13459:
13451:
13445:
13441:
13440:
13432:
13424:
13418:
13414:
13413:
13405:
13397:
13391:
13387:
13386:
13378:
13370:
13364:
13360:
13359:
13351:
13349:
13340:
13334:
13330:
13329:
13321:
13313:
13307:
13303:
13302:
13294:
13286:
13280:
13276:
13275:
13267:
13259:
13253:
13249:
13248:
13243:
13237:
13229:
13223:
13219:
13218:
13210:
13202:
13196:
13192:
13191:
13183:
13181:
13172:
13166:
13162:
13161:
13153:
13145:
13139:
13135:
13134:
13126:
13118:
13114:
13107:
13099:
13095:
13091:
13087:
13083:
13079:
13075:
13071:
13064:
13056:
13050:
13046:
13045:
13037:
13029:
13023:
13019:
13018:
13010:
13002:
12996:
12992:
12991:
12983:
12975:
12969:
12965:
12964:
12956:
12954:
12945:
12939:
12935:
12934:
12926:
12918:
12912:
12908:
12904:
12901:. p. 1.
12900:
12893:
12891:
12889:
12880:
12874:
12870:
12869:
12861:
12853:
12847:
12843:
12842:
12834:
12826:
12820:
12816:
12815:
12807:
12799:
12793:
12789:
12788:
12780:
12772:
12766:
12762:
12761:
12753:
12745:
12739:
12735:
12734:
12726:
12718:
12712:
12708:
12707:
12699:
12697:
12688:
12682:
12678:
12677:
12669:
12667:
12658:
12652:
12648:
12647:
12639:
12637:
12628:
12622:
12618:
12617:
12609:
12601:
12595:
12591:
12590:
12581:
12579:
12570:
12564:
12560:
12559:
12551:
12544:
12539:
12537:
12528:
12522:
12518:
12517:
12509:
12501:
12495:
12491:
12490:
12482:
12474:
12468:
12464:
12463:
12455:
12453:
12444:
12438:
12434:
12433:
12425:
12417:
12412:
12411:
12402:
12394:
12388:
12384:
12383:
12375:
12373:
12364:
12358:
12354:
12353:
12345:
12343:
12341:
12339:
12337:
12328:
12322:
12318:
12317:
12309:
12307:
12305:
12303:
12294:
12288:
12284:
12283:
12275:
12267:
12259:
12255:
12254:
12248:
12241:
12233:
12229:
12225:
12221:
12217:
12213:
12209:
12205:
12198:
12190:
12186:
12182:
12178:
12174:
12170:
12166:
12162:
12155:
12147:
12143:
12138:
12133:
12128:
12123:
12119:
12115:
12111:
12107:
12103:
12096:
12094:
12092:
12090:
12081:
12075:
12071:
12070:
12062:
12054:
12048:
12044:
12043:
12035:
12027:
12019:
12015:
12014:
12008:
12001:
11999:
11990:
11984:
11980:
11979:
11974:
11968:
11966:
11964:
11955:
11949:
11945:
11944:
11936:
11934:
11925:
11919:
11915:
11914:
11906:
11904:
11902:
11900:
11898:
11889:
11883:
11879:
11878:
11870:
11868:
11866:
11857:
11851:
11847:
11846:
11838:
11836:
11827:
11821:
11817:
11816:
11808:
11806:
11797:
11791:
11787:
11786:
11778:
11770:
11764:
11760:
11759:
11751:
11749:
11740:
11734:
11730:
11729:
11721:
11719:
11717:
11715:
11713:
11711:
11702:
11696:
11692:
11691:
11683:
11681:
11672:
11668:
11661:
11659:
11657:
11655:
11653:
11651:
11649:
11647:
11645:
11643:
11634:
11628:
11624:
11623:
11615:
11613:
11611:
11609:
11600:
11594:
11590:
11589:
11584:
11578:
11570:
11564:
11560:
11559:
11551:
11543:
11537:
11533:
11532:
11527:
11526:Sylvie Roelly
11520:
11518:
11509:
11505:
11498:
11490:
11484:
11480:
11479:
11471:
11463:
11457:
11453:
11452:
11444:
11436:
11430:
11426:
11425:
11417:
11409:
11403:
11399:
11398:
11390:
11382:
11376:
11372:
11371:
11363:
11361:
11352:
11346:
11342:
11341:
11333:
11325:
11319:
11315:
11314:
11306:
11298:
11292:
11288:
11287:
11279:
11271:
11265:
11261:
11260:
11252:
11244:
11238:
11234:
11233:
11225:
11217:
11211:
11207:
11206:
11198:
11190:
11184:
11180:
11179:
11171:
11163:
11157:
11153:
11152:
11144:
11136:
11130:
11126:
11125:
11117:
11115:
11113:
11111:
11109:
11107:
11105:
11103:
11094:
11088:
11084:
11083:
11075:
11073:
11071:
11069:
11060:
11054:
11050:
11049:
11041:
11039:
11037:
11035:
11033:
11031:
11029:
11027:
11025:
11023:
11021:
11019:
11017:
11015:
11013:
11011:
11002:
10996:
10992:
10991:
10983:
10981:
10979:
10977:
10975:
10973:
10964:
10958:
10954:
10953:
10945:
10937:
10931:
10927:
10926:
10921:
10914:
10906:
10902:
10898:
10894:
10890:
10886:
10882:
10878:
10871:
10869:
10867:
10865:
10856:
10850:
10846:
10845:
10837:
10829:
10825:
10821:
10817:
10813:
10809:
10805:
10801:
10797:
10793:
10786:
10784:
10782:
10780:
10778:
10776:
10774:
10772:
10763:
10759:
10755:
10749:
10745:
10741:
10736:
10731:
10727:
10720:
10718:
10716:
10714:
10712:
10710:
10708:
10706:
10704:
10694:
10693:
10684:
10676:
10670:
10666:
10665:
10657:
10649:
10643:
10639:
10635:
10634:
10626:
10618:
10612:
10608:
10604:
10603:
10595:
10593:
10584:
10578:
10574:
10570:
10569:
10561:
10553:
10547:
10543:
10539:
10538:
10530:
10522:
10516:
10512:
10508:
10507:
10499:
10491:
10485:
10481:
10477:
10476:
10468:
10460:
10458:1-886529-03-5
10454:
10450:
10449:
10444:
10438:
10430:
10424:
10420:
10416:
10415:
10407:
10399:
10393:
10389:
10385:
10384:
10376:
10368:
10362:
10358:
10354:
10353:
10345:
10337:
10331:
10327:
10323:
10322:
10314:
10306:
10300:
10296:
10292:
10291:
10283:
10275:
10269:
10265:
10264:
10259:
10253:
10245:
10239:
10235:
10234:
10226:
10224:
10222:
10220:
10218:
10216:
10214:
10212:
10210:
10208:
10206:
10204:
10195:
10189:
10185:
10184:
10176:
10174:
10172:
10170:
10168:
10159:
10153:
10149:
10148:
10140:
10138:
10136:
10127:
10121:
10117:
10116:
10108:
10106:
10104:
10102:
10093:
10092:
10084:
10082:
10080:
10078:
10076:
10074:
10072:
10070:
10068:
10063:
10047:
10038:
10029:
10020:
10011:
10002:
9993:
9984:
9974:
9967:
9963:
9959:
9955:
9951:
9947:
9941:
9937:
9926:
9923:
9921:
9918:
9916:
9913:
9911:
9908:
9906:
9903:
9901:
9898:
9896:
9893:
9891:
9888:
9886:
9883:
9881:
9878:
9876:
9873:
9871:
9868:
9866:
9863:
9860:
9857:
9855:
9852:
9850:
9847:
9845:
9842:
9840:
9837:
9836:
9829:
9815:
9805:
9803:
9799:
9794:
9792:
9782:
9779:
9765:
9745:
9737:
9717:
9714:
9711:
9703:
9687:
9679:
9664:
9661:
9658:
9650:
9634:
9626:
9611:
9608:
9605:
9597:
9581:
9573:
9572:
9571:
9557:
9537:
9527:
9518:
9516:
9511:
9506:
9497:
9495:
9491:
9486:
9482:
9472:
9470:
9466:
9462:
9457:
9453:
9451:
9447:
9442:
9438:
9434:
9430:
9426:
9425:finite groups
9421:
9419:
9415:
9412:, written by
9411:
9410:Eugene Onegin
9407:
9402:
9401:Andrey Markov
9392:
9389:
9388:Harry Bateman
9385:
9381:
9376:
9373:
9370:in 1909 when
9369:
9364:
9362:
9358:
9354:
9350:
9340:
9338:
9337:Percy Daniell
9334:
9329:
9327:
9323:
9320:, known as a
9319:
9315:
9311:
9306:
9304:
9299:
9297:
9293:
9289:
9285:
9277:
9273:
9269:
9265:
9261:
9257:
9255:
9250:
9246:
9236:
9233:
9229:
9213:
9204:
9202:
9197:
9192:
9188:
9178:
9176:
9172:
9161:
9152:
9149:
9147:
9146:Martin Hairer
9143:
9139:
9138:Fields Medals
9135:
9131:
9127:
9123:
9117:
9115:
9111:
9107:
9103:
9099:
9094:
9091:
9087:
9083:
9078:
9076:
9072:
9068:
9064:
9050:
9046:
9041:
9037:
9035:
9031:
9027:
9023:
9022:Harald Cramér
9019:
9015:
9011:
9007:
9003:
8998:
8996:
8986:
8984:
8980:
8976:
8972:
8968:
8963:
8961:
8957:
8953:
8949:
8945:
8941:
8940:David Hilbert
8937:
8933:
8923:
8921:
8917:
8913:
8909:
8904:
8900:
8896:
8886:
8884:
8880:
8876:
8872:
8868:
8864:
8860:
8855:
8853:
8849:
8845:
8841:
8840:Blaise Pascal
8837:
8836:Pierre Fermat
8821:
8807:
8798:
8796:
8792:
8788:
8787:point process
8772:
8762:
8761:Point process
8755:Point process
8752:
8738:
8728:
8718:
8716:
8715:subordinators
8694:
8691:
8685:
8682:
8674:
8669:
8653:
8646:
8643:
8640:
8636:
8631:
8627:
8620:
8616:
8611:
8607:
8604:
8601:
8594:
8590:
8585:
8581:
8574:
8570:
8565:
8556:
8542:
8539:
8536:
8514:
8510:
8506:
8503:
8500:
8495:
8491:
8487:
8484:
8464:
8444:
8436:
8429:
8419:
8415:
8413:
8407:
8405:
8399:
8385:
8382:
8377:
8373:
8369:
8364:
8360:
8356:
8351:
8347:
8343:
8338:
8334:
8330:
8325:
8321:
8317:
8312:
8308:
8287:
8284:
8279:
8275:
8271:
8266:
8262:
8258:
8253:
8249:
8240:
8236:
8231:
8226:
8216:
8202:
8193:
8191:
8187:
8182:
8180:
8179:Kai Lai Chung
8176:
8171:
8167:
8162:
8160:
8156:
8151:
8149:
8143:
8128:
8125:
8115:
8111:
8094:
8091:
8088:
8078:
8077:unit interval
8059:
8056:
8053:
8047:
8027:
8007:
7999:
7995:
7973:
7970:
7944:
7941:
7938:
7927:
7923:
7917:
7890:
7886:
7882:
7877:
7873:
7865:
7862:
7854:
7850:
7846:
7841:
7837:
7830:
7797:
7792:
7788:
7784:
7780:
7776:
7771:
7767:
7763:
7755:
7754:
7753:
7725:
7721:
7714:
7703:
7699:
7692:
7686:
7680:
7672:
7668:
7664:
7659:
7655:
7648:
7623:
7606:
7601:
7597:
7593:
7571:
7566:
7562:
7558:
7546:Orthogonality
7543:
7529:
7509:
7478:
7474:
7470:
7465:
7461:
7453:
7450:
7442:
7438:
7434:
7429:
7425:
7418:
7385:
7380:
7376:
7372:
7368:
7364:
7359:
7355:
7351:
7343:
7342:
7341:
7326:
7321:
7312:
7308:
7299:
7295:
7291:
7283:
7279:
7272:
7268:
7263:
7254:
7250:
7241:
7237:
7233:
7225:
7221:
7214:
7210:
7205:
7201:
7195:
7187:
7183:
7179:
7174:
7170:
7163:
7138:
7121:
7116:
7112:
7108:
7086:
7081:
7077:
7073:
7058:
7043:
7034:
7030:
7023:
7020:
7017:
7014:
7006:
7002:
6995:
6991:
6969:
6960:
6956:
6949:
6946:
6943:
6940:
6932:
6928:
6921:
6917:
6896:
6893:
6888:
6884:
6880:
6877:
6874:
6869:
6865:
6839:
6836:
6828:
6812:
6789:
6786:
6776:
6750:
6730:
6716:
6714:
6709:
6695:
6673:
6645:
6642:
6639:
6631:
6628:
6623:
6619:
6592:
6589:
6586:
6583:
6580:
6572:
6569:
6564:
6560:
6529:
6523:
6517:
6514:
6510:
6507:
6487:
6484:
6481:
6461:
6458:
6450:
6439:
6416:
6411:
6386:
6383:
6380:
6360:
6337:
6334:
6324:
6298:
6289:
6287:
6282:
6273:
6259:
6239:
6231:
6215:
6195:
6173:
6170:
6167:
6161:
6158:
6155:
6145:
6141:
6137:
6132:
6128:
6121:
6113:
6111:
6095:
6075:
6052:
6049:
6039:
6013:
5993:
5979:
5977:
5976:metric spaces
5961:
5952:
5947:
5944:
5939:
5937:
5933:
5915:
5912:
5909:
5901:
5897:
5893:
5888:
5884:
5877:
5869:
5855:
5852:
5849:
5829:
5809:
5786:
5783:
5773:
5747:
5727:
5707:
5699:
5689:
5653:
5622:
5618:
5597:
5575:
5546:
5526:
5523:
5520:
5498:
5494:
5473:
5453:
5433:
5430:
5427:
5424:
5421:
5401:
5398:
5395:
5368:
5363:
5351:
5346:
5314:
5311:
5301:
5273:
5270:
5267:
5257:
5237:
5233:
5223:
5221:
5217:
5213:
5209:
5193:
5190:
5187:
5167:
5145:
5142:
5139:
5135:
5112:
5108:
5087:
5084:
5081:
5061:
5041:
5031:
5016:
5007:
5005:
4987:
4980:
4976:
4971:
4967:
4964:
4957:
4953:
4948:
4939:
4925:
4903:
4899:
4895:
4892:
4889:
4884:
4880:
4859:
4837:
4833:
4812:
4809:
4806:
4786:
4778:
4773:
4763:
4747:
4730:
4727:
4714:
4704:
4700:
4695:
4691:
4688:
4685:
4679:
4669:
4665:
4660:
4649:
4643:
4640:
4623:
4620:
4614:
4604:
4600:
4596:
4593:
4590:
4585:
4581:
4576:
4567:
4553:
4533:
4530:
4527:
4524:
4521:
4518:
4513:
4509:
4501:
4485:
4465:
4456:
4442:
4422:
4393:
4389:
4381:
4378:
4375:
4372:
4363:
4359:
4351:
4324:
4320:
4316:
4313:
4310:
4307:
4302:
4298:
4293:
4285:This measure
4270:
4265:
4262:
4248:
4244:
4236:
4233:
4230:
4227:
4218:
4214:
4206:
4200:
4197:
4194:
4187:
4183:
4179:
4176:
4173:
4168:
4164:
4159:
4150:
4136:
4133:
4128:
4124:
4120:
4117:
4114:
4109:
4105:
4096:
4080:
4060:
4050:
4040:
4038:
4034:
4030:
4012:
4003:
4000:
3994:
3988:
3985:
3979:
3976:
3967:
3964:
3958:
3952:
3944:
3930:
3908:
3902:
3899:
3893:
3887:
3884:
3878:
3875:
3869:
3863:
3855:
3852:
3848:
3839:
3825:
3803:
3799:
3778:
3769:
3755:
3752:
3749:
3729:
3707:
3703:
3682:
3660:
3656:
3633:
3630:
3626:
3605:
3585:
3563:
3558:
3555:
3551:
3547:
3544:
3541:
3538:
3530:
3528:
3527:image measure
3512:
3504:
3485:
3482:
3472:
3444:
3440:
3430:
3427:
3408:
3406:
3405:Banach spaces
3390:
3370:
3350:
3326:
3322:
3317:
3313:
3306:
3302:
3297:
3274:
3270:
3266:
3261:
3257:
3230:
3227:
3221:
3216:
3212:
3185:
3182:
3176:
3171:
3167:
3140:
3137:
3131:
3128:
3108:
3085:
3082:
3079:
3076:
3070:
3064:
3052:
3042:
3040:
3036:
3035:path function
3032:
3016:
2996:
2973:
2970:
2947:
2944:
2941:
2938:
2932:
2929:
2926:
2920:
2909:
2893:
2885:
2867:
2864:
2858:
2855:
2849:
2846:
2843:
2837:
2829:
2828:
2809:
2806:
2783:
2780:
2777:
2774:
2768:
2765:
2762:
2756:
2745:
2741:
2731:
2729:
2726:-dimensional
2713:
2705:
2701:
2697:
2681:
2674:
2664:
2650:
2647:
2644:
2624:
2602:
2587:
2571:
2551:
2543:
2539:
2535:
2534:parameter set
2531:
2515:
2501:
2487:
2467:
2460:from the set
2459:
2441:
2437:
2414:
2410:
2400:
2383:
2380:
2360:
2357:
2354:
2331:
2328:
2325:
2322:
2316:
2313:
2310:
2304:
2281:
2258:
2252:
2232:
2229:
2226:
2204:
2198:
2195:
2192:
2189:
2183:
2177:
2166:
2152:
2126:
2123:
2097:
2094:
2084:
2069:
2035:
2027:
2011:
1991:
1983:
1967:
1959:
1943:
1911:
1872:
1869:
1859:
1846:
1831:
1817:
1808:
1794:
1784:
1780:
1775:
1765:
1761:
1759:
1755:
1751:
1750:Almost surely
1747:
1733:
1713:
1704:
1688:
1666:
1662:
1660:
1656:
1652:
1648:
1644:
1638:
1628:
1614:
1611:
1608:
1588:
1585:
1582:
1562:
1554:
1549:
1546:
1542:
1537:
1527:
1525:
1509:
1489:
1486:
1483:
1463:
1455:
1451:
1445:
1430:
1413:
1410:
1407:
1404:
1399:
1395:
1365:
1362:
1356:
1353:
1333:
1311:
1307:
1283:
1277:
1269:
1250:
1244:
1224:
1217:or simply as
1198:
1192:
1165:
1161:
1135:
1132:
1129:
1119:
1115:
1089:
1086:
1083:
1072:
1066:
1049:
1047:
1029:
1002:
994:
990:
986:
981:
976:
973:
969:
959:
957:
953:
949:
944:
939:
937:
933:
929:
925:
921:
917:
913:
909:
905:
901:
897:
893:
883:
881:
865:
857:
841:
821:
813:
809:
805:
801:
796:
794:
790:
789:
784:
783:discrete-time
780:
779:
774:
773:
772:discrete time
766:
764:
751:
747:
742:
738:
736:
732:
728:
724:
708:
700:
697:known as the
696:
692:
688:
684:
680:
669:
667:
663:
659:
655:
654:real analysis
651:
647:
643:
639:
635:
631:
627:
623:
619:
615:
611:
607:
603:
598:
596:
592:
588:
584:
580:
576:
572:
568:
564:
560:
556:
551:
548:
544:
540:
536:
532:
527:
525:
521:
517:
513:
509:
505:
501:
497:
493:
489:
485:
481:
477:
473:
470:
466:
465:thermal noise
462:
458:
454:
450:
446:
442:
438:
434:
430:
426:
422:
416:
377:
373:
365:
361:
356:
345:
340:
338:
333:
331:
326:
325:
323:
322:
317:
314:
312:
309:
308:
307:
306:
301:
298:
296:
293:
291:
288:
286:
283:
281:
278:
276:
273:
272:
271:
270:
265:
262:
260:
257:
255:
252:
250:
247:
246:
245:
244:
237:
234:
232:
229:
227:
224:
222:
219:
217:
214:
212:
209:
207:
204:
202:
199:
198:
197:
194:
192:
189:
185:
182:
180:
177:
175:
172:
170:
167:
165:
162:
160:
157:
156:
155:
152:
148:
145:
144:
143:
140:
136:
133:
131:
128:
126:
123:
121:
118:
116:
113:
112:
111:
108:
106:
103:
101:
98:
97:
96:
95:
90:
87:
85:
84:Indeterminism
82:
78:
75:
74:
73:
70:
66:
63:
62:
61:
58:
57:
56:
55:
51:
47:
46:
43:
40:
39:
36:
32:
31:
19:
22337: /
22333: /
22323:
22199:Optimization
22162:Superalgebra
22021:Field theory
21994:Mathematical
21972: /
21969:
21820:Chaos theory
21803:Gauge theory
21731:Graph theory
21626:Cryptography
21426:Econometrics
21388:Wiener space
21276:ItĂŽ integral
21177:Inequalities
21066:Self-similar
21036:GaussâMarkov
21026:Exchangeable
21006:CĂ dlĂ g paths
20942:Risk process
20894:LIBOR market
20763:Random graph
20758:Random field
20570:Superprocess
20508:LĂ©vy process
20503:Jump process
20478:Hunt process
20314:Markov chain
20272:
20213:
20188:
20164:
20147:. Springer.
20143:
20119:
20108:
20084:
20063:
20042:
20025:
20021:
19986:
19953:
19949:
19922:
19918:
19905:
19901:
19877:
19868:
19856:
19811:
19804:
19784:
19777:
19757:
19750:
19730:
19700:
19693:
19673:
19666:
19646:
19639:
19619:
19612:
19592:
19585:
19565:
19558:
19538:
19531:
19511:
19479:
19472:
19447:
19443:
19436:
19416:
19409:
19376:
19372:
19366:
19346:
19339:
19319:
19312:
19285:
19279:
19246:
19242:
19236:
19222:(2): 92â96.
19219:
19215:
19186:
19152:
19115:
19111:
19105:
19070:
19064:
19044:
19037:
19004:
19000:
18994:
18961:
18957:
18951:
18911:
18907:
18845:
18841:
18806:
18796:
18768:
18755:
18722:
18718:
18676:
18672:
18666:
18657:
18647:
18627:
18620:
18600:
18593:
18573:
18566:
18546:
18539:
18519:
18512:
18492:
18485:
18465:
18435:
18428:
18408:
18401:
18392:
18388:
18382:
18373:
18369:
18363:
18343:
18336:
18327:
18323:
18317:
18292:
18288:
18282:
18255:
18251:
18241:
18221:
18214:
18194:
18187:
18167:
18160:
18151:
18147:
18141:
18132:
18128:
18086:
18082:
18018:
18014:
17976:
17972:
17962:
17935:
17931:
17891:
17855:
17849:
17824:
17820:
17776:(1): 15â16.
17773:
17769:
17763:
17738:
17734:
17728:
17698:(1): 31â32.
17695:
17691:
17685:
17652:
17648:
17642:
17612:(1): 22â23.
17609:
17605:
17599:
17572:
17568:
17506:
17502:
17496:
17463:
17459:
17425:
17421:
17374:(1): 28â35.
17371:
17367:
17338:
17331:
17311:
17304:
17284:
17277:
17250:
17246:
17236:
17216:
17209:
17182:
17150:
17143:
17110:
17106:
17100:
17080:
17073:
17053:
17046:
17026:
17019:
16999:
16992:
16972:
16965:
16941:
16902:
16895:
16875:
16868:
16848:
16841:
16821:
16814:
16794:
16787:
16767:
16760:
16740:
16733:
16724:
16720:
16689:
16655:
16648:
16628:
16621:
16580:math/0609294
16570:
16566:
16560:
16540:
16533:
16513:
16506:
16486:
16479:
16459:
16452:
16432:
16402:
16395:
16375:
16368:
16348:
16314:
16284:
16277:
16267:
16260:
16240:
16208:
16176:
16169:
16149:
16142:
16122:
16115:
16095:
16088:
16068:
16061:
16041:
16034:
16014:
16007:
15987:
15980:
15960:
15953:
15933:
15926:
15906:
15899:
15879:
15849:
15819:
15812:
15792:
15785:
15765:
15758:
15738:
15731:
15711:
15704:
15684:
15677:
15657:
15650:
15630:
15623:
15603:
15596:
15576:
15569:
15549:
15517:
15487:
15455:
15406:
15370:
15365:
15345:
15338:
15328:
15321:
15309:
15289:
15259:
15229:
15222:
15202:
15172:
15165:
15145:
15107:
15093:
15073:
15066:
15046:
15039:
15019:
15012:
14992:
14985:
14965:
14935:
14905:
14898:
14878:
14871:
14851:
14821:
14814:
14794:
14787:
14767:
14735:
14728:
14708:
14688:
14658:
14651:
14631:
14597:
14590:
14570:
14536:
14529:
14509:
14502:
14482:
14475:
14455:
14448:
14428:
14421:
14401:
14362:
14355:
14335:
14328:
14308:
14301:
14281:
14274:
14254:
14247:
14227:
14188:
14184:
14174:
14154:
14147:
14127:
14094:
14087:
14067:
14037:
14030:
14010:
13980:
13955:
13947:
13922:
13914:
13894:
13887:
13867:
13860:
13840:
13833:
13813:
13806:
13786:
13779:
13759:
13752:
13732:
13700:
13693:
13673:
13666:
13646:
13639:
13619:
13612:
13603:
13599:
13593:
13577:. Springer.
13573:
13566:
13546:
13539:
13519:
13512:
13492:
13485:
13465:
13458:
13438:
13431:
13411:
13404:
13384:
13377:
13357:
13327:
13320:
13300:
13293:
13273:
13266:
13246:
13242:Daniel Revuz
13236:
13216:
13209:
13189:
13159:
13152:
13132:
13125:
13116:
13112:
13106:
13073:
13069:
13063:
13043:
13036:
13016:
13009:
12989:
12982:
12962:
12932:
12925:
12898:
12867:
12860:
12840:
12833:
12813:
12806:
12786:
12779:
12759:
12752:
12732:
12725:
12705:
12675:
12645:
12615:
12608:
12588:
12557:
12550:
12515:
12508:
12488:
12481:
12461:
12431:
12424:
12409:
12401:
12381:
12351:
12315:
12281:
12274:
12251:
12240:
12207:
12203:
12197:
12164:
12160:
12154:
12109:
12105:
12068:
12061:
12041:
12034:
12011:
12007:"Stochastic"
11977:
11942:
11912:
11876:
11844:
11814:
11784:
11777:
11757:
11727:
11689:
11670:
11666:
11621:
11587:
11577:
11557:
11550:
11530:
11507:
11503:
11497:
11477:
11470:
11454:. Springer.
11450:
11443:
11423:
11416:
11396:
11389:
11369:
11339:
11332:
11312:
11305:
11285:
11278:
11258:
11251:
11231:
11224:
11204:
11197:
11177:
11170:
11150:
11143:
11123:
11081:
11047:
10989:
10951:
10944:
10924:
10913:
10880:
10876:
10843:
10836:
10795:
10791:
10725:
10691:
10683:
10663:
10656:
10632:
10625:
10601:
10567:
10560:
10536:
10529:
10505:
10498:
10474:
10467:
10447:
10437:
10413:
10406:
10382:
10375:
10351:
10344:
10320:
10313:
10289:
10282:
10266:. Springer.
10262:
10252:
10232:
10182:
10146:
10114:
10090:
10046:
10037:
10028:
10019:
10010:
10001:
9992:
9983:
9973:
9957:
9953:
9949:
9945:
9940:
9890:Random field
9880:Markov chain
9859:Entropy rate
9806:
9795:
9788:
9780:
9735:
9733:
9528:
9524:
9514:
9507:
9503:
9478:
9454:
9422:
9398:
9377:
9365:
9359:published a
9346:
9330:
9307:
9300:
9288:Paris Bourse
9281:
9242:
9232:George PĂłlya
9205:
9195:
9190:
9187:Karl Pearson
9184:
9181:Random walks
9174:
9167:
9158:
9150:
9118:
9097:
9095:
9090:Gilbert Hunt
9079:
9059:
9048:
9026:World War II
8999:
8994:
8992:
8964:
8929:
8912:Josiah Gibbs
8892:
8856:
8847:
8832:
8799:
8794:
8790:
8786:
8764:
8730:
8727:Random field
8721:Random field
8673:Banach space
8670:
8667:
8434:
8431:
8428:LĂ©vy process
8422:LĂ©vy process
8416:
8408:
8403:
8400:
8232:
8228:
8194:
8183:
8163:
8155:random walks
8152:
8145:
8142:Markov chain
8123:
8121:
8112:
7993:
7925:
7921:
7919:
7621:
7549:
7501:
7137:uncorrelated
7136:
7064:
6826:
6825:are said be
6722:
6719:Independence
6710:
6290:
6281:Separability
6280:
6279:
6276:Separability
6187:
6109:
6108:are said be
5985:
5948:
5942:
5940:
5935:
5931:
5929:
5698:modification
5697:
5695:
5692:Modification
5229:
5219:
5215:
5211:
5032:
5008:
5001:
4777:Stationarity
4776:
4775:
4766:Stationarity
4761:
4457:
4284:
4094:
4052:
4037:distribution
4036:
4032:
4028:
4026:
3922:
3770:
3577:
3502:
3419:
3050:
3048:
3038:
3034:
3030:
2907:
2883:
2881:
2742:is a single
2739:
2737:
2695:
2670:
2533:
2529:
2507:
2401:
2218:
2070:
1910:sample space
1842:
1809:
1785:
1781:
1777:
1762:
1748:
1679:
1661:in liquids.
1640:
1552:
1550:
1541:Random walks
1539:
1447:
1055:
993:random field
992:
988:
984:
979:
977:
971:
967:
965:
951:
947:
942:
940:
931:
923:
919:
907:
891:
889:
879:
855:
811:
807:
803:
799:
797:
792:
786:
782:
776:
770:
767:
760:
749:
745:
734:
730:
722:
698:
675:
672:Introduction
602:random walks
599:
595:random field
566:
562:
561:. The terms
554:
552:
547:A. K. Erlang
539:Paris Bourse
528:
488:neuroscience
439:, where the
420:
375:
369:
316:Tree diagram
311:Venn diagram
275:Independence
235:
221:Markov chain
105:Sample space
22339:topics list
22273:Mathematics
22189:Game theory
22090:Topological
22056:Topological
22051:Statistical
22014:Hamiltonian
21471:Ruin theory
21409:Disciplines
21281:ItĂŽ's lemma
21056:Predictable
20731:Percolation
20714:Potts model
20709:Ising model
20673:White noise
20631:Differences
20493:ItĂŽ process
20433:Cox process
20329:Loop-erased
20324:Random walk
19007:(1): 1â36.
18679:(1): 1â20.
18021:(5): 1655.
16727:(11): 1336.
16344:C. C. Heyde
13606:(11): 1341.
13119:(11): 1338.
11673:(11): 1337.
9791:Joseph Doob
9384:Hans Geiger
9372:A.K. Erlang
9333:Jean Perrin
9191:random walk
9134:rough paths
9045:Joseph Doob
9002:Joseph Doob
8956:Ămile Borel
8555:increments
8175:Joseph Doob
8166:state space
7620:are called
7135:are called
6829:if for all
6827:independent
6713:Joseph Doob
5842:if for all
5236:total order
2908:sample path
2884:realization
2696:state space
2667:State space
2586:total order
1834:Definitions
1536:Random walk
1530:Random walk
962:Terminology
928:Joseph Doob
763:cardinality
750:realization
735:realization
699:state space
630:probability
606:martingales
231:Random walk
72:Determinism
60:Probability
22354:Categories
22245:Psychology
22209:Statistics
22009:Lagrangian
21636:Statistics
21572:Algorithms
21481:Statistics
21261:Filtration
21162:Kolmogorov
21146:Blumenthal
21071:Stationary
21011:Continuous
20999:Properties
20884:HullâWhite
20626:Martingale
20513:Local time
20401:Fractional
20379:pure birth
20028:(1): 1â42.
19855:" section
18135:(1): 1â42.
17938:(1): 251.
17466:(2): 128.
17422:Biometrika
17179:Seneta, E.
17107:Biometrika
16933:Cox, D. R.
13076:(1): 1â2.
12268:required.)
12028:required.)
10058:References
9895:Randomness
9494:Kiyosi ItĂŽ
9114:Jean Ville
9063:Kiyosi ItĂŽ
8875:Carl Gauss
8219:Martingale
8124:regularity
8118:Regularity
7622:orthogonal
5936:equivalent
5232:filtration
5226:Filtration
3249:such that
3031:trajectory
2704:real lines
2026:measurable
1643:stationary
950:pre-dates
920:stochastik
892:stochastic
778:continuous
642:set theory
545:, used by
541:, and the
449:Stochastic
376:stochastic
142:Experiment
89:Randomness
35:statistics
22250:Sociology
22240:Chemistry
22036:Effective
22031:Conformal
22026:Classical
21898:Geometric
21871:Geometric
21393:Classical
20406:Geometric
20396:Excursion
20013:0749-2170
19970:0306-7734
19941:0091-1798
19871:July 2018
19464:0024-6093
19450:(1): 57.
19401:0002-9505
19263:0306-7734
19132:0346-1238
19097:0169-7161
19029:117623580
19021:0003-9519
18986:117623580
18978:0003-9519
18964:(1): 25.
18936:154003579
18928:0967-2567
18862:0960-1627
18739:0306-7734
18693:0306-7734
18309:0346-1238
18295:(1): 98.
18274:0091-1798
18105:0021-9002
18053:0091-1798
18028:0909.4213
17995:0091-1798
17954:0021-9002
17841:0024-6093
17827:(1): 33.
17798:117623580
17790:0003-9519
17755:0003-3790
17720:189764116
17712:0003-9519
17677:120059181
17669:0003-9519
17634:189764116
17626:0003-9519
17591:0091-1798
17523:0002-9890
17488:118011380
17480:0306-7734
17442:0006-3444
17388:0002-9890
17269:0315-0860
17127:0006-3444
16605:0883-4237
16342:P. Hall;
14207:117171116
13098:117623580
13090:0003-9519
12232:119439925
12224:0025-5831
12189:122842868
12181:0025-5831
10897:0306-7734
10828:125163415
10812:0025-5572
10762:0749-2170
10730:CiteSeerX
10511:CRC Press
9966:Skorokhod
9944:The term
9715:∈
9662:∈
9609:∈
9570:include:
9378:In 1910,
9308:In 1905,
9185:In 1905,
9177:in 1713.
9148:in 2014.
9110:Paul LĂ©vy
9071:integrals
9014:Paul LĂ©vy
8960:Paul LĂ©vy
8938:in 1900,
8698:∞
8644:−
8628:−
8605:…
8582:−
8540:−
8507:≤
8504:⋯
8501:≤
8488:≤
8386:…
8288:…
7977:∞
7870:∀
7831:
7809:⟺
7735:¯
7687:
7649:
7458:∀
7419:
7397:⟺
7296:μ
7292:−
7238:μ
7234:−
7202:
7164:
7018:…
6944:…
6894:∈
6878:…
6840:∈
6774:Ω
6670:Ω
6643:∈
6629:∈
6590:∩
6584:∈
6570:∈
6533:∞
6527:∞
6524:−
6511:⊂
6485:⊂
6447:Ω
6420:Ω
6417:⊂
6408:Ω
6384:⊂
6322:Ω
6159:∈
6037:Ω
5853:∈
5771:Ω
5676:Ω
5524:∈
5454:≤
5431:∈
5399:≤
5369:⊆
5352:⊆
5299:Ω
5271:∈
5191:∈
5085:∈
4968:…
4893:…
4810:∈
4728:∈
4715:ω
4689:…
4680:ω
4647:Ω
4644:∈
4641:ω
4594:…
4577:μ
4531:×
4528:⋯
4525:×
4423:μ
4376:…
4294:μ
4263:−
4231:…
4201:∘
4177:…
4160:μ
4134:∈
4118:…
4081:μ
4073:with law
4035:, or the
4001:∈
3995:ω
3983:Ω
3980:∈
3977:ω
3953:μ
3900:∈
3894:ω
3882:Ω
3879:∈
3876:ω
3853:−
3753:∈
3631:−
3606:∘
3556:−
3548:∘
3539:μ
3470:Ω
3437:→
3434:Ω
3431::
3314:−
3267:≤
3234:∞
3222:∈
3189:∞
3177:∈
3144:∞
3083:∈
3051:increment
3045:Increment
2977:Ω
2974:∈
2971:ω
2945:∈
2933:ω
2862:→
2850:ω
2844:⋅
2813:Ω
2810:∈
2807:ω
2781:∈
2769:ω
2648:∈
2538:real line
2530:index set
2504:Index set
2458:functions
2387:Ω
2384:∈
2381:ω
2358:∈
2329:∈
2317:ω
2230:∈
2196:∈
2130:Σ
2082:Ω
2056:Σ
2048:-algebra
2036:σ
1944:σ
1896:Ω
1857:Ω
1734:μ
1714:μ
1649:that are
1586:−
1487:−
1411:≥
1369:∞
1133:∈
1087:∈
978:The term
890:The word
886:Etymology
723:increment
687:real line
679:index set
583:real line
553:The term
480:chemistry
457:bacterial
135:Singleton
22325:Category
21974:analysis
21893:Exterior
21866:Exterior
21746:Analysis
21708:Discrete
21582:analysis
21514:Category
21398:Abstract
20932:BĂŒhlmann
20538:Compound
20210:(1962).
20186:(2015).
20141:(2010).
20112:. Wiley.
20106:(1990).
20046:. SIAM.
19894:Articles
18940:Archived
18874:Archived
18870:14422885
18785:Archived
18061:17288507
16939:(1980).
16613:62552177
16346:(2014).
15103:Marc Yor
14125:(2013).
12247:"Random"
12210:(1): 1.
12146:16587907
11975:(2006).
11585:(2014).
11528:(2011).
11373:. SIAM.
11289:. SIAM.
10445:(1996).
10295:Elsevier
10260:(2014).
9832:See also
9228:lattices
8235:sequence
5414:, where
5388:for all
5208:Khinchin
5180:for all
3695:, where
2700:integers
2508:The set
1888:, where
1433:Examples
1052:Notation
1046:manifold
800:discrete
727:outcomes
652:such as
646:topology
634:calculus
579:interval
575:integers
472:molecule
216:Variance
22335:outline
22266:Related
22235:Biology
22085:Bosonic
22046:Quantum
21996:physics
21962: (
21694:Solvers
21021:Ergodic
20909:VaĆĄĂÄek
20751:Poisson
20411:Meander
19381:Bibcode
19271:1403518
18747:1402616
18701:1403034
18033:Bibcode
17531:2974673
17396:2589523
17135:2333419
16585:Bibcode
15436:, p. 24
15316:, p. 22
12137:1076423
12114:Bibcode
10820:3621649
9962:Gikhman
9368:Denmark
9032:to the
8930:At the
8824:History
8791:process
6232:, then
5943:version
4478:of the
2827:mapping
2744:outcome
1958:algebra
896:English
685:of the
581:of the
524:finance
492:physics
484:ecology
476:biology
130:Outcome
21908:Vector
21903:Tensor
21881:Vector
21876:Tensor
21577:design
21361:Tanaka
21046:Mixing
21041:Markov
20914:Wilkie
20879:HoâLee
20874:Heston
20646:Super-
20391:Bridge
20339:Biased
20196:
20172:
20151:
20127:
20092:
20071:
20050:
20011:
20001:
19976:
19968:
19939:
19851:This "
19819:
19792:
19765:
19738:
19708:
19681:
19654:
19627:
19600:
19573:
19546:
19519:
19487:
19462:
19424:
19399:
19354:
19327:
19300:
19269:
19261:
19194:
19164:
19160:â466.
19130:
19095:
19085:
19052:
19027:
19019:
18984:
18976:
18934:
18926:
18868:
18860:
18745:
18737:
18699:
18691:
18635:
18608:
18581:
18554:
18527:
18500:
18473:
18443:
18416:
18351:
18307:
18272:
18229:
18202:
18175:
18103:
18059:
18051:
17993:
17952:
17906:
17870:
17839:
17796:
17788:
17753:
17718:
17710:
17675:
17667:
17632:
17624:
17589:
17529:
17521:
17486:
17478:
17440:
17394:
17386:
17346:
17319:
17292:
17267:
17224:
17197:
17158:
17133:
17125:
17088:
17061:
17034:
17007:
16980:
16953:
16910:
16883:
16856:
16829:
16802:
16775:
16748:
16697:
16663:
16636:
16611:
16603:
16548:
16521:
16494:
16467:
16440:
16410:
16383:
16356:
16322:
16292:
16248:
16216:
16184:
16157:
16130:
16103:
16076:
16049:
16022:
15995:
15968:
15941:
15914:
15887:
15857:
15827:
15800:
15773:
15746:
15719:
15692:
15665:
15638:
15611:
15584:
15557:
15525:
15495:
15463:
15414:
15353:
15297:
15267:
15237:
15210:
15180:
15153:
15115:
15081:
15054:
15027:
15000:
14973:
14943:
14913:
14886:
14859:
14829:
14802:
14775:
14743:
14716:
14666:
14639:
14605:
14578:
14544:
14517:
14490:
14463:
14436:
14409:
14370:
14343:
14316:
14289:
14262:
14235:
14205:
14162:
14135:
14102:
14075:
14045:
14018:
13988:
13935:
13902:
13875:
13848:
13821:
13794:
13767:
13740:
13708:
13681:
13654:
13627:
13581:
13554:
13527:
13500:
13473:
13446:
13419:
13392:
13365:
13335:
13308:
13281:
13254:
13224:
13197:
13167:
13140:
13096:
13088:
13051:
13024:
12997:
12970:
12940:
12913:
12875:
12848:
12821:
12794:
12767:
12740:
12713:
12683:
12653:
12623:
12596:
12565:
12545:, p. 1
12523:
12496:
12469:
12439:
12389:
12359:
12323:
12289:
12230:
12222:
12187:
12179:
12144:
12134:
12076:
12049:
11985:
11950:
11920:
11884:
11852:
11822:
11792:
11765:
11735:
11697:
11629:
11595:
11565:
11538:
11485:
11458:
11431:
11404:
11377:
11347:
11320:
11293:
11266:
11239:
11212:
11185:
11158:
11131:
11089:
11055:
10997:
10959:
10932:
10903:
10895:
10851:
10826:
10818:
10810:
10760:
10750:
10732:
10671:
10644:
10613:
10579:
10548:
10517:
10486:
10455:
10425:
10394:
10363:
10332:
10301:
10270:
10240:
10190:
10154:
10122:
9700:is an
9361:thesis
9274:, and
9030:Sweden
9020:, and
8973:, and
8948:axioms
8861:wrote
4498:-fold
4093:, its
3838:gives
3578:where
3501:, the
2825:, the
1960:, and
943:random
683:subset
664:, and
644:, and
624:, and
577:or an
514:, and
429:family
360:Wiener
77:System
65:Axioms
22041:Gauge
21214:Tools
20990:M/M/c
20985:M/M/1
20980:M/G/1
20970:Fluid
20636:Local
20034:Books
19978:80836
19974:S2CID
19267:JSTOR
19025:S2CID
18982:S2CID
18943:(PDF)
18932:S2CID
18904:(PDF)
18877:(PDF)
18866:S2CID
18838:(PDF)
18788:(PDF)
18765:(PDF)
18743:JSTOR
18697:JSTOR
18057:S2CID
18023:arXiv
17794:S2CID
17716:S2CID
17673:S2CID
17630:S2CID
17527:JSTOR
17484:S2CID
17392:JSTOR
17131:JSTOR
16609:S2CID
16575:arXiv
14203:S2CID
14191:(4).
13094:S2CID
12262:
12228:S2CID
12185:S2CID
12022:
10905:80836
10901:S2CID
10824:S2CID
10816:JSTOR
10544:Inc.
9932:Notes
9677:; and
9647:is a
9594:is a
8936:Paris
3343:is a
1980:is a
1936:is a
1908:is a
1044:or a
900:Greek
810:or a
441:index
435:in a
423:is a
419:) or
110:Event
21166:LĂ©vy
20965:Bulk
20849:Chen
20641:Sub-
20599:Both
20194:ISBN
20170:ISBN
20149:ISBN
20125:ISBN
20090:ISBN
20069:ISBN
20048:ISBN
20009:ISSN
19999:ISBN
19966:ISSN
19937:ISSN
19817:ISBN
19790:ISBN
19763:ISBN
19736:ISBN
19706:ISBN
19679:ISBN
19652:ISBN
19625:ISBN
19598:ISBN
19571:ISBN
19544:ISBN
19517:ISBN
19485:ISBN
19460:ISSN
19422:ISBN
19397:ISSN
19352:ISBN
19325:ISBN
19298:ISBN
19259:ISSN
19192:ISBN
19162:ISBN
19128:ISSN
19116:1969
19093:ISSN
19083:ISBN
19050:ISBN
19017:ISSN
18974:ISSN
18924:ISSN
18858:ISSN
18735:ISSN
18689:ISSN
18633:ISBN
18606:ISBN
18579:ISBN
18552:ISBN
18525:ISBN
18498:ISBN
18471:ISBN
18441:ISBN
18414:ISBN
18349:ISBN
18305:ISSN
18293:1995
18270:ISSN
18227:ISBN
18200:ISBN
18173:ISBN
18101:ISSN
18049:ISSN
17991:ISSN
17950:ISSN
17904:ISBN
17868:ISBN
17837:ISSN
17786:ISSN
17751:ISSN
17708:ISSN
17665:ISSN
17622:ISSN
17587:ISSN
17519:ISSN
17476:ISSN
17438:ISSN
17384:ISSN
17344:ISBN
17317:ISBN
17290:ISBN
17265:ISSN
17222:ISBN
17195:ISBN
17156:ISBN
17123:ISSN
17086:ISBN
17059:ISBN
17032:ISBN
17005:ISBN
16978:ISBN
16951:ISBN
16947:p. 3
16908:ISBN
16881:ISBN
16854:ISBN
16827:ISBN
16800:ISBN
16773:ISBN
16746:ISBN
16695:ISBN
16661:ISBN
16634:ISBN
16601:ISSN
16546:ISBN
16519:ISBN
16492:ISBN
16465:ISBN
16438:ISBN
16408:ISBN
16381:ISBN
16354:ISBN
16320:ISBN
16290:ISBN
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