6420:
5611:
6415:{\displaystyle {\begin{aligned}{\boldsymbol {\pi }}^{(k)}&=\mathbf {x} \left(\mathbf {U\Sigma U} ^{-1}\right)\left(\mathbf {U\Sigma U} ^{-1}\right)\cdots \left(\mathbf {U\Sigma U} ^{-1}\right)\\&=\mathbf {xU\Sigma } ^{k}\mathbf {U} ^{-1}\\&=\left(a_{1}\mathbf {u} _{1}^{\mathsf {T}}+a_{2}\mathbf {u} _{2}^{\mathsf {T}}+\cdots +a_{n}\mathbf {u} _{n}^{\mathsf {T}}\right)\mathbf {U\Sigma } ^{k}\mathbf {U} ^{-1}\\&=a_{1}\lambda _{1}^{k}\mathbf {u} _{1}^{\mathsf {T}}+a_{2}\lambda _{2}^{k}\mathbf {u} _{2}^{\mathsf {T}}+\cdots +a_{n}\lambda _{n}^{k}\mathbf {u} _{n}^{\mathsf {T}}&&u_{i}\bot u_{j}{\text{ for }}i\neq j\\&=\lambda _{1}^{k}\left\{a_{1}\mathbf {u} _{1}^{\mathsf {T}}+a_{2}\left({\frac {\lambda _{2}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{2}^{\mathsf {T}}+a_{3}\left({\frac {\lambda _{3}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{3}^{\mathsf {T}}+\cdots +a_{n}\left({\frac {\lambda _{n}}{\lambda _{1}}}\right)^{k}\mathbf {u} _{n}^{\mathsf {T}}\right\}\end{aligned}}}
2868:
9492:, whenever probabilities are used to represent unknown or unmodelled details of the system, if it can be assumed that the dynamics are time-invariant, and that no relevant history need be considered which is not already included in the state description. For example, a thermodynamic state operates under a probability distribution that is difficult or expensive to acquire. Therefore, Markov Chain Monte Carlo method can be used to draw samples randomly from a black-box to approximate the probability distribution of attributes over a range of objects.
58:
9924:
31:
481:
2600:
14430:] Extensive, wide-ranging book meant for specialists, written for both theoretical computer scientists as well as electrical engineers. With detailed explanations of state minimization techniques, FSMs, Turing machines, Markov processes, and undecidability. Excellent treatment of Markov processes pp. 449ff. Discusses Z-transforms, D transforms in their context.
4576:
608:). Moreover, the time index need not necessarily be real-valued; like with the state space, there are conceivable processes that move through index sets with other mathematical constructs. Notice that the general state space continuous-time Markov chain is general to such a degree that it has no designated term.
2257:
10630:
and position of the runners. Mark Pankin shows that Markov chain models can be used to evaluate runs created for both individual players as well as a team. He also discusses various kinds of strategies and play conditions: how Markov chain models have been used to analyze statistics for game situations such as
9395:, that every aperiodic and irreducible Markov chain is isomorphic to a Bernoulli scheme; thus, one might equally claim that Markov chains are a "special case" of Bernoulli schemes. The isomorphism generally requires a complicated recoding. The isomorphism theorem is even a bit stronger: it states that
9658:
towards a desired class of compounds such as drugs or natural products. As a molecule is grown, a fragment is selected from the nascent molecule as the "current" state. It is not aware of its past (that is, it is not aware of what is already bonded to it). It then transitions to the next state when a
9634:
A reaction network is a chemical system involving multiple reactions and chemical species. The simplest stochastic models of such networks treat the system as a continuous time Markov chain with the state being the number of molecules of each species and with reactions modeled as possible transitions
655:
Since the system changes randomly, it is generally impossible to predict with certainty the state of a Markov chain at a given point in the future. However, the statistical properties of the system's future can be predicted. In many applications, it is these statistical properties that are important.
639:
A discrete-time random process involves a system which is in a certain state at each step, with the state changing randomly between steps. The steps are often thought of as moments in time, but they can equally well refer to physical distance or any other discrete measurement. Formally, the steps are
10620:
Usually musical systems need to enforce specific control constraints on the finite-length sequences they generate, but control constraints are not compatible with Markov models, since they induce long-range dependencies that violate the Markov hypothesis of limited memory. In order to overcome this
798:
where, at each step, the position may change by +1 or â1 with equal probability. From any position there are two possible transitions, to the next or previous integer. The transition probabilities depend only on the current position, not on the manner in which the position was reached. For example,
10629:
Markov chain models have been used in advanced baseball analysis since 1960, although their use is still rare. Each half-inning of a baseball game fits the Markov chain state when the number of runners and outs are considered. During any at-bat, there are 24 possible combinations of number of outs
635:
describing the probabilities of particular transitions, and an initial state (or initial distribution) across the state space. By convention, we assume all possible states and transitions have been included in the definition of the process, so there is always a next state, and the process does not
9807:
by modeling texts in a natural language (such as
English) as generated by an ergodic Markov process, where each letter may depend statistically on previous letters. Such idealized models can capture many of the statistical regularities of systems. Even without describing the full structure of the
9765:
applications. Solar irradiance variability at any location over time is mainly a consequence of the deterministic variability of the sun's path across the sky dome and the variability in cloudiness. The variability of accessible solar irradiance on Earth's surface has been modeled using Markov
9639:
of molecules in solution in state A, each of which can undergo a chemical reaction to state B with a certain average rate. Perhaps the molecule is an enzyme, and the states refer to how it is folded. The state of any single enzyme follows a Markov chain, and since the molecules are essentially
4029:
of 1. If there is more than one unit eigenvector then a weighted sum of the corresponding stationary states is also a stationary state. But for a Markov chain one is usually more interested in a stationary state that is the limit of the sequence of distributions for some initial distribution.
680:
to hold. 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 vector of values, so proving a weak law of large numbers without the independence assumption, which had been commonly
10194:
Markov models have also been used to analyze web navigation behavior of users. A user's web link transition on a particular website can be modeled using first- or second-order Markov models and can be used to make predictions regarding future navigation and to personalize the web page for an
748:, 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
1404:
is not possible. After the second draw, the third draw depends on which coins have so far been drawn, but no longer only on the coins that were drawn for the first state (since probabilistically important information has since been added to the scenario). In this way, the likelihood of the
791:, which are considered the most important and central stochastic processes in the theory of stochastic processes. These two processes are Markov processes in continuous time, while random walks on the integers and the gambler's ruin problem are examples of Markov processes in discrete time.
8614:
9383:
is a special case of a Markov chain where the transition probability matrix has identical rows, which means that the next state is independent of even the current state (in addition to being independent of the past states). A Bernoulli scheme with only two possible states is known as a
4443:
8892:
9666:
may be modeled using Markov chains. Based on the reactivity ratios of the monomers that make up the growing polymer chain, the chain's composition may be calculated (for example, whether monomers tend to add in alternating fashion or in long runs of the same monomer). Due to
3634:
9259:
9622:
8410:
is the time, starting in a given set of states until the chain arrives in a given state or set of states. The distribution of such a time period has a phase type distribution. The simplest such distribution is that of a single exponentially distributed transition.
2206:
4989:
whose each row sums to 1. So it needs any nĂn independent linear equations of the (nĂn+n) equations to solve for the nĂn variables. In this example, the n equations from âQ multiplied by the right-most column of (P-In)â have been replaced by the n stochastic
4449:
9322:
Markov models are used to model changing systems. There are 4 main types of models, that generalize Markov chains depending on whether every sequential state is observable or not, and whether the system is to be adjusted on the basis of observations made:
12811:
Kopp, V. S.; Kaganer, V. M.; Schwarzkopf, J.; Waidick, F.; Remmele, T.; Kwasniewski, A.; Schmidbauer, M. (2011). "X-ray diffraction from nonperiodic layered structures with correlations: Analytical calculation and experiment on mixed
Aurivillius films".
1744:
508:"): it is a process for which predictions can be made regarding future outcomes based solely on its present state andâmost importantlyâsuch predictions are just as good as the ones that could be made knowing the process's full history. This means that,
5551:
740:'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,
5347:
595:
Note that there is no definitive agreement in the literature on the use of some of the terms that signify special cases of Markov processes. Usually the term "Markov chain" is reserved for a process with a discrete set of times, that is, a
7956:
is the set of all states for the Markov chain. Let the sigma-algebra on the probability space be generated by the cylinder sets. Let the probability measure be generated by the stationary distribution, and the Markov chain transition. Let
802:
A series of independent states (for example, a series of coin flips) satisfies the formal definition of a Markov chain. However, the theory is usually applied only when the probability distribution of the next state depends on the current
9651:, can be viewed as a Markov chain, where at each time step the reaction proceeds in some direction. While Michaelis-Menten is fairly straightforward, far more complicated reaction networks can also be modeled with Markov chains.
2595:{\displaystyle {\begin{aligned}{}&\Pr(X_{n}=x_{n}\mid X_{n-1}=x_{n-1},X_{n-2}=x_{n-2},\dots ,X_{1}=x_{1})\\=&\Pr(X_{n}=x_{n}\mid X_{n-1}=x_{n-1},X_{n-2}=x_{n-2},\dots ,X_{n-m}=x_{n-m}){\text{ for }}n>m\end{aligned}}}
11676:
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)".
11496:
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)".
527:
with a countable state space (thus regardless of the nature of time), but it is also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space).
812:
Suppose that there is a coin purse containing five quarters (each worth 25Âą), five dimes (each worth 10Âą), and five nickels (each worth 5Âą), and one by one, coins are randomly drawn from the purse and are set on a table. If
4774:
540:
and time parameter index need to be specified. The following table gives an overview of the different instances of Markov processes for different levels of state space generality and for discrete time v. continuous time:
667:
studied Markov processes in the early 20th century, publishing his first paper on the topic in 1906. Markov
Processes in continuous time were discovered long before his work in the early 20th century in the form of the
1259:
possible states, where each state represents the number of coins of each type (from 0 to 5) that are on the table. (Not all of these states are reachable within 6 draws.) Suppose that the first draw results in state
8481:
4584:
Because there are a number of different special cases to consider, the process of finding this limit if it exists can be a lengthy task. However, there are many techniques that can assist in finding this limit. Let
5075:
4219:
1983:
5356:
is a row stochastic matrix, its largest left eigenvalue is 1. If there is a unique stationary distribution, then the largest eigenvalue and the corresponding eigenvector is unique too (because there is no other
4335:
3169:
8768:
2778:
6986:
778:
problem are examples of Markov processes. Some variations of these processes were studied hundreds of years earlier in the context of independent variables. Two important examples of Markov processes are the
10682:. The Markov chain forecasting models utilize a variety of settings, from discretizing the time series, to hidden Markov models combined with wavelets, and the Markov chain mixture distribution model (MCM).
10134:
9980:
8969:
4651:
6779:
6554:
735:
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
720:
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
7030:
otherwise. Periodicity, transience, recurrence and positive and null recurrence are class properties â that is, if one state has the property then all states in its communicating class have the property.
6638:
with each other if both are reachable from one another by a sequence of transitions that have positive probability. This is an equivalence relation which yields a set of communicating classes. A class is
1999:
10360:
for each note is constructed, completing a transition probability matrix (see below). An algorithm is constructed to produce output note values based on the transition matrix weightings, which could be
10328:", for example, are represented exactly by Markov chains. At each turn, the player starts in a given state (on a given square) and from there has fixed odds of moving to certain other states (squares).
3395:
799:
the transition probabilities from 5 to 4 and 5 to 6 are both 0.5, and all other transition probabilities from 5 are 0. These probabilities are independent of whether the system was previously in 4 or 6.
1835:
631:
The changes of state of the system are called transitions. The probabilities associated with various state changes are called transition probabilities. The process is characterized by a state space, a
7540:
9144:
7050:
Since periodicity is a class property, if a Markov chain is irreducible, then all its states have the same period. In particular, if one state is aperiodic, then the whole Markov chain is aperiodic.
4977:
10296:. In current research, it is common to use a Markov chain to model how once a country reaches a specific level of economic development, the configuration of structural factors, such as size of the
5220:
10223:
Markov chains are used in finance and economics to model a variety of different phenomena, including the distribution of income, the size distribution of firms, asset prices and market crashes.
9876:
initiated the subject in 1917. This makes them critical for optimizing the performance of telecommunications networks, where messages must often compete for limited resources (such as bandwidth).
10610:
th-order chains tend to "group" particular notes together, while 'breaking off' into other patterns and sequences occasionally. These higher-order chains tend to generate results with a sense of
8369:
8067:
2216:. Every stationary chain can be proved to be time-homogeneous by Bayes' rule.A necessary and sufficient condition for a time-homogeneous Markov chain to be stationary is that the distribution of
8486:
5616:
2262:
3969:
3837:
4327:
4275:
615:
of a Markov chain does not have any generally agreed-on restrictions: the term may refer to a process on an arbitrary state space. However, many applications of Markov chains employ finite or
12338:
10203:
Markov chain methods have also become very important for generating sequences of random numbers to accurately reflect very complicated desired probability distributions, via a process called
9511:
12776:
Kutchukian, Peter S.; Lou, David; Shakhnovich, Eugene I. (2009-06-15). "FOG: Fragment
Optimized Growth Algorithm for the de Novo Generation of Molecules Occupying Druglike Chemical Space".
9635:
of the chain. Markov chains and continuous-time Markov processes are useful in chemistry when physical systems closely approximate the Markov property. For example, imagine a large number
8102:
7934:
4571:{\displaystyle {\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}}
11354:
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".
4692:
922:
619:
state spaces, which have a more straightforward statistical analysis. Besides time-index and state-space parameters, there are many other variations, extensions and generalizations (see
6611:
523:
or a 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
10251:
and Adlai J. Fisher, which builds upon the convenience of earlier regime-switching models. It uses an arbitrarily large Markov chain to drive the level of volatility of asset returns.
8248:
In some cases, apparently non-Markovian processes may still have
Markovian representations, constructed by expanding the concept of the "current" and "future" states. For example, let
3033:
604:
without explicit mention. In addition, there are other extensions of Markov processes that are referred to as such but do not necessarily fall within any of these four categories (see
4113:
652:
for the system at the next step (and in fact at all future steps) depends only on the current state of the system, and not additionally on the state of the system at previous steps.
10169:
8686:
3905:
8703:
states that the necessary and sufficient condition for a process to be reversible is that the product of transition rates around a closed loop must be the same in both directions.
1554:
1257:
10617:
Markov chains can be used structurally, as in
Xenakis's Analogique A and B. Markov chains are also used in systems which use a Markov model to react interactively to music input.
10243:(1989), who used a Markov chain to model switches between periods of high and low GDP growth (or, alternatively, economic expansions and recessions). A more recent example is the
7987:
5454:
2856:
are chosen such that each row of the transition rate matrix sums to zero, while the row-sums of a probability transition matrix in a (discrete) Markov chain are all equal to one.
14189:
Munkhammar, J.; van der Meer, D.W.; Widén, J. (2019). "Probabilistic forecasting of high-resolution clear-sky index time-series using a Markov-chain mixture distribution model".
9054:
7862:
4060:
7670:
7393:
5446:
1499:
1028:
14607:
Original paper by A.A Markov (1913): An Example of Statistical Investigation of the Text Eugene Onegin Concerning the Connection of Samples in Chains (translated from Russian)
12971:
Aguiar, R. J.; Collares-Pereira, M.; Conde, J. P. (1988). "Simple procedure for generating sequences of daily radiation values using a library of Markov transition matrices".
9131:
7468:
4015:
3709:
1221:
could be defined to represent the state where there is one quarter, zero dimes, and five nickels on the table after 6 one-by-one draws. This new model could be represented by
7776:
965:
12741:
Kutchukian, Peter; Lou, David; Shakhnovich, Eugene (2009). "FOG: Fragment Optimized Growth Algorithm for the de Novo Generation of Molecules occupying Druglike Chemical".
8207:
7108:
5231:
4066:
and its eigenvectors have their relative proportions preserved. Since the components of π are positive and the constraint that their sum is unity can be rewritten as
3199:
14581:
8181:
7627:
1448:
1402:
1303:
1219:
13987:
8461:
7729:
2984:
9097:
9077:
9019:
2871:
The continuous time Markov chain is characterized by the transition rates, the derivatives with respect to time of the transition probabilities between states i and j.
382:
of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs
10189:
8227:
7954:
7589:
2945:
2670:
2637:
875:
7183:
It can be shown that a finite state irreducible Markov chain is ergodic if it has an aperiodic state. More generally, a Markov chain is ergodic if there is a number
6812:
3237:
10079:
7697:
7333:
7266:
7178:
7024:
2900:
2241:
1357:
1330:
1170:
1139:
1109:
1082:
1055:
992:
838:
14679:
994:, but the earlier values as well, then we can determine which coins have been drawn, and we know that the next coin will not be a nickel; so we can determine that
6838:
9283:
Another discrete-time process that may be derived from a continuous-time Markov chain is a ÎŽ-skeleton—the (discrete-time) Markov chain formed by observing
14235:
A. A. Markov (1971). "Extension of the limit theorems of probability theory to a sum of variables connected in a chain". reprinted in Appendix B of: R. Howard.
10052:
10032:
10012:
8136:
7882:
7796:
7563:
7488:
7416:
7353:
7306:
7239:
7151:
4277:
is found, then the stationary distribution of the Markov chain in question can be easily determined for any starting distribution, as will be explained below.
1057:
we might guess that we had drawn four dimes and two nickels, in which case it would certainly be possible to draw another nickel next. Thus, our guesses about
9465:
9461:
9750:", arranging these chains in several recursive layers ("wafering") and producing more efficient test setsâsamplesâas a replacement for exhaustive testing.
1868:
7206:
Some authors call any irreducible, positive recurrent Markov chains ergodic, even periodic ones. In fact, merely irreducible Markov chains correspond to
7198:
A Markov chain with more than one state and just one out-going transition per state is either not irreducible or not aperiodic, hence cannot be ergodic.
3050:
15214:
349:
12142:
4707:
13340:
15038:
9835:
Markov chains are also the basis for hidden Markov models, which are an important tool in such diverse fields as telephone networks (which use the
8111:
Markov chains with finite state spaces have a unique stationary distribution, the above construction is unambiguous for irreducible Markov chains.
4853:
and substitutes each of its elements by one, and on the other one substitutes the corresponding element (the one in the same column) in the vector
8232:
The terminology is inconsistent. Given a Markov chain with a stationary distribution that is strictly positive on all states, the Markov chain is
9453:
8609:{\displaystyle {\begin{aligned}k_{i}^{A}=0&{\text{ for }}i\in A\\-\sum _{j\in S}q_{ij}k_{j}^{A}=1&{\text{ for }}i\notin A.\end{aligned}}}
15641:
14634:
14331:. Grundlehren der mathematischen Wissenschaften. Vol. I (121). Translated by Fabius, Jaap; Greenberg, Vida Lazarus; Maitra, Ashok Prasad;
14269:
11990:
Schmitt, Florian; Rothlauf, Franz (2001). "On the Importance of the Second Largest Eigenvalue on the Convergence Rate of Genetic Algorithms".
10254:
Dynamic macroeconomics makes heavy use of Markov chains. An example is using Markov chains to exogenously model prices of equity (stock) in a
15171:
15151:
10276:
arguments, where current structural configurations condition future outcomes. An example is the reformulation of the idea, originally due to
13663:
Calvet, Laurent; Adlai Fisher (2004). "How to Forecast long-run volatility: regime-switching and the estimation of multifractal processes".
13006:
Ngoko, B. O.; Sugihara, H.; Funaki, T. (2014). "Synthetic generation of high temporal resolution solar radiation data using Markov models".
5042:
4438:{\displaystyle \mathbf {P} ={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\qquad \mathbf {P} ^{2k}=I\qquad \mathbf {P} ^{2k+1}=\mathbf {P} }
14486:. 2nd rev. ed., 1981, XVI, 288 p., Softcover Springer Series in Statistics. (Originally published by Allen & Unwin Ltd., London, 1973)
9360:
8887:{\displaystyle s_{ij}={\begin{cases}{\frac {q_{ij}}{\sum _{k\neq i}q_{ik}}}&{\text{if }}i\neq j\\0&{\text{otherwise}}.\end{cases}}}
7191:. In case of a fully connected transition matrix, where all transitions have a non-zero probability, this condition is fulfilled with
4168:
14244:
Markov, A. A. (2006). "An Example of Statistical Investigation of the Text Eugene Onegin Concerning the Connection of Samples in Chains".
11311:
Jarrow, Robert; Protter, Philip (2004). "A short history of stochastic integration and mathematical finance: The early years, 1880â1970".
15555:
14418:
13919:
13701:
6610:
Considering a collection of Markov chains whose evolution takes in account the state of other Markov chains, is related to the notion of
7133:
If all states in an irreducible Markov chain are ergodic, then the chain is said to be ergodic. Equivalently, there exists some integer
2683:
13729:
6885:
9746:
Several theorists have proposed the idea of the Markov chain statistical test (MCST), a method of conjoining Markov chains to form a "
9476:
Markov chains have been employed in a wide range of topics across the natural and social sciences, and in technological applications.
7053:
If a finite Markov chain is irreducible, then all states are positive recurrent, and it has a unique stationary distribution given by
15472:
12203:
10084:
9930:
9847:
8907:
6585:
681:
regarded as a requirement for such mathematical laws to hold. Markov later used Markov chains to study the distribution of vowels in
563:
14313:
6690:
6490:
4600:
3629:{\displaystyle \Pr(X_{t_{n+1}}=i_{n+1}\mid X_{t_{0}}=i_{0},X_{t_{1}}=i_{1},\ldots ,X_{t_{n}}=i_{n})=p_{i_{n}i_{n+1}}(t_{n+1}-t_{n})}
15156:
10239:
and random walk models were popular in the literature of the 1960s. Regime-switching models of business cycles were popularized by
17:
623:). For simplicity, most of this article concentrates on the discrete-time, discrete state-space case, unless mentioned otherwise.
15482:
15166:
13319:
9254:{\displaystyle \pi ={-\varphi (\operatorname {diag} (Q))^{-1} \over \left\|\varphi (\operatorname {diag} (Q))^{-1}\right\|_{1}}.}
11967:
1753:
15524:
15421:
14578:
9795:
7493:
342:
34:
A diagram representing a two-state Markov process. The numbers are the probability of changing from one state to another state.
14606:
13991:
12427:. Probability and its applications (2. ed., ed.). New York, NY Berlin Heidelberg: Springer. Proposition 8.6 (page 145).
10235:
was the first to observe that stock prices followed a random walk. The random walk was later seen as evidence in favor of the
6602:
The use of Markov chains in Markov chain Monte Carlo methods covers cases where the process follows a continuous state space.
4882:
15711:
15701:
15547:
15239:
15224:
14545:
14491:
14414:
14381:
14352:
14041:
13862:
13381:
12626:
12432:
12374:
12322:
11951:
11879:
11835:
11808:
11748:
11718:
11660:
11598:
11571:
11475:
11338:
11295:
11233:
11201:
11066:
11039:
11012:
10985:
10958:
10861:
10834:
10807:
5176:
5015:
th row or column is otherwise filled with 0's, then that row or column will remain unchanged in all of the subsequent powers
1548:, namely that the probability of moving to the next state depends only on the present state and not on the previous states:
15611:
15575:
9734:
5121:
be the matrix of eigenvectors (each normalized to having an L2 norm equal to 1) where each column is a left eigenvector of
649:
9617:{\displaystyle {\ce {{E}+{\underset {Substrate \atop binding}{S<=>E}}{\overset {Catalytic \atop step}{S->E}}+P}}}
8270:
7992:
15879:
15616:
9629:. The enzyme (E) binds a substrate (S) and produces a product (P). Each reaction is a state transition in a Markov chain.
8712:
7898:
6595:
Many results for Markov chains with finite state space can be generalized to chains with uncountable state space through
3920:
3756:
15528:
14726:
14627:
13585:
Hamilton, James (1989). "A new approach to the economic analysis of nonstationary time series and the business cycle".
11781:
690:
13043:"Stochastic generation of synthetic minutely irradiance time series derived from mean hourly weather observation data"
6570:
is the dominant term. The smaller the ratio is, the faster the convergence is. Random noise in the state distribution
4287:
4235:
2822:
with dimensions equal to that of the state space and initial probability distribution defined on the state space. For
729:
became interested in Markov chains, eventually resulting in him publishing in 1938 a detailed study on Markov chains.
15681:
14530:
14523:
Performance and reliability analysis of computer systems: an example-based approach using the SHARPE software package
14508:
14477:
14472:
E. Nummelin. "General irreducible Markov chains and non-negative operators". Cambridge University Press, 1984, 2004.
14467:
14309:
14294:
14276:
13941:
13152:
Munkhammar, J.; Widén, J. (2018). "A Markov-chain probability distribution mixture approach to the clear-sky index".
12659:
12490:
12264:
12183:
12152:
11915:
11152:
11131:
11110:
11090:
10702:
9659:
fragment is attached to it. The transition probabilities are trained on databases of authentic classes of compounds.
3653:
2201:{\displaystyle \Pr(X_{0}=x_{0},X_{1}=x_{1},\ldots ,X_{k}=x_{k})=\Pr(X_{n}=x_{0},X_{n+1}=x_{1},\ldots ,X_{n+k}=x_{k})}
513:
335:
323:
282:
8072:
7904:
5102:
linearly independent eigenvectors, speed of convergence is elaborated as follows. (For non-diagonalizable, that is,
15726:
15532:
15516:
15431:
15259:
15229:
14651:
12642:
Anderson, David F.; Kurtz, Thomas G. (2011), "Continuous Time Markov Chain Models for Chemical Reaction Networks",
3866:
is a (row) vector, whose entries are non-negative and sum to 1, is unchanged by the operation of transition matrix
672:. Markov was interested in studying an extension of independent random sequences, motivated by a disagreement with
15920:
15631:
15596:
15565:
15560:
14996:
14913:
10692:
10231:
and co-author Charles Bonini used a Markov chain model to derive a stationary Yule distribution of firm sizes.
4661:
931:
To see why this is the case, suppose that in the first six draws, all five nickels and a quarter are drawn. Thus
880:
745:
524:
213:
149:
7335:
are positive. The exponent is purely a graph-theoretic property, since it depends only on whether each entry of
704:
with an aim to study card shuffling. Other early uses of Markov chains include a diffusion model, introduced by
15570:
15199:
15194:
15001:
14898:
14006:
13886:
13082:
Munkhammar, J.; Widén, J. (2018). "An N-state Markov-chain mixture distribution model of the clear-sky index".
12916:"Comparison of Parameter Estimation Methods in Stochastic Chemical Kinetic Models: Examples in Systems Biology"
10920:
9915:(job service times are exponentially distributed) and describe completed services (departures) from the queue.
2989:
741:
261:
122:
1739:{\displaystyle \Pr(X_{n+1}=x\mid X_{1}=x_{1},X_{2}=x_{2},\ldots ,X_{n}=x_{n})=\Pr(X_{n+1}=x\mid X_{n}=x_{n}),}
15884:
15661:
15497:
15396:
15381:
14920:
14793:
14709:
14620:
14571:
13898:
10762:
10244:
9648:
9399:
5546:{\displaystyle \mathbf {x} ^{\mathsf {T}}=\sum _{i=1}^{n}a_{i}\mathbf {u} _{i},\qquad a_{i}\in \mathbb {R} .}
582:
15656:
15536:
10139:
9778:
8636:
4069:
3876:
722:
15666:
14442:(1st ed.). Englewood Cliffs, NJ: Prentice-Hall, Inc. Library of Congress Card Catalog Number 59-12841.
9392:
6619:
4229:
2790:
1224:
399:
15671:
15307:
7960:
4829:(see the definition above). It is sometimes sufficient to use the matrix equation above and the fact that
15269:
14853:
14798:
14714:
14566:
14428:(1st ed.). New York, NY: John Wiley and Sons, Inc. Library of Congress Card Catalog Number 67-25924.
14120:
13337:
10772:
10651:
10236:
9027:
7805:
7187:
such that any state can be reached from any other state in any number of steps less or equal to a number
6615:
4036:
15601:
7635:
7358:
15606:
15591:
15234:
15204:
14771:
14669:
14596:
14228:
A. A. Markov (1906) "Rasprostranenie zakona bol'shih chisel na velichiny, zavisyaschie drug ot druga".
10707:
5564:
from right and continue this operation with the results, in the end we get the stationary distribution
5419:
4982:
1515:
1453:
997:
391:
117:
14061:
10674:
Markov chains have been used for forecasting in several areas: for example, price trends, wind power,
10666:, and Academias Neutronium). Several open-source text generation libraries using Markov chains exist.
9103:
7425:
15905:
15686:
15487:
15401:
15386:
15317:
14893:
14776:
14674:
10884:
10727:
10717:
9626:
8700:
5342:{\displaystyle 1=|\lambda _{1}|>|\lambda _{2}|\geq |\lambda _{3}|\geq \cdots \geq |\lambda _{n}|.}
3977:
765:
677:
233:
13783:
13677:
13599:
9766:
chains, also including modeling the two states of clear and cloudiness as a two-state Markov chain.
8793:
7398:
There are several combinatorial results about the exponent when there are finitely many states. Let
15520:
15406:
14908:
14883:
14828:
14320:
13459:
13426:
13398:
13324:
13226:
Thomsen, Samuel W. (2009), "Some evidence concerning the genesis of Shannon's information theory",
12000:
11321:
10767:
10712:
10697:
10204:
9716:
9654:
An algorithm based on a Markov chain was also used to focus the fragment-based growth of chemicals
7737:
3299:
934:
422:
418:
417:
of real-world processes. They provide the basis for general stochastic simulation methods known as
292:
287:
176:
161:
14561:
12231:
8186:
7056:
4857:, and next left-multiplies this latter vector by the inverse of transformed former matrix to find
3174:
15821:
15811:
15626:
15502:
15284:
15209:
15023:
14888:
14744:
14699:
13636:
12109:
10889:
10722:
10337:
9698:
9449:
9411:
9355:
8751:
8731:. Strictly speaking, the EMC is a regular discrete-time Markov chain, sometimes referred to as a
8401:
8141:
7885:
7594:
6669:
4150:
If the Markov chain is irreducible and aperiodic, then there is a unique stationary distribution
3851:
3662:
1747:
1408:
1362:
1263:
1179:
509:
271:
142:
13314:
12060:
Franzke, Brandon; Kosko, Bart (1 October 2011). "Noise can speed convergence in Markov chains".
9927:
A state diagram that represents the PageRank algorithm with a transitional probability of M, or
15910:
15763:
15691:
15116:
15106:
14950:
13778:
13672:
13594:
13454:
13421:
11995:
11316:
10212:
9829:
8434:
7702:
2950:
2816:
166:
14158:
13916:
13712:
13371:
13254:"An alignment-free method to find and visualise rearrangements between pairs of DNA sequences"
9082:
9062:
9004:
6614:. This corresponds to the situation when the state space has a (Cartesian-) product form. See
648:, and the random process is a mapping of these to states. The Markov property states that the
15915:
15786:
15768:
15748:
15743:
15462:
15294:
15274:
15121:
15064:
14903:
14813:
13826:
K McAlpine; E Miranda; S Hoggar (1999). "Making Music with Algorithms: A Case-Study System".
11708:
10897:
10264:
produce annual tables of the transition probabilities for bonds of different credit ratings.
10174:
9489:
9433:
8212:
7939:
7568:
3743:
2917:
2642:
2609:
847:
669:
307:
266:
171:
137:
14601:
14436:
13740:
11650:
6791:
3212:
15861:
15816:
15806:
15492:
15467:
15436:
15416:
15254:
15176:
15161:
15028:
14198:
13770:
13413:
13265:
13161:
13126:
13091:
13054:
13015:
12980:
12927:
12868:
12821:
12697:
12069:
11623:
10752:
10675:
10289:
10261:
10057:
8386:
7799:
7675:
7311:
7244:
7156:
7002:
3731:
2878:
2219:
1335:
1308:
1148:
1117:
1087:
1060:
1033:
1030:
with probability 1. But if we do not know the earlier values, then based only on the value
970:
816:
717:
297:
191:
84:
2867:
8:
15856:
15696:
15621:
15426:
15186:
15096:
14986:
14332:
13310:
12532:
10757:
10742:
10255:
10014:
in the stationary distribution on the following Markov chain on all (known) webpages. If
9873:
9825:
9804:
9774:
9712:
9671:, second-order Markov effects may also play a role in the growth of some polymer chains.
9345:
6817:
426:
256:
198:
186:
181:
14501:
Probability and Statistics with Reliability, Queueing, and Computer Science Applications
14202:
13774:
13448:
13417:
13269:
13165:
13130:
13095:
13058:
13019:
12984:
12931:
12872:
12825:
12701:
12073:
11627:
11419:
Seneta, E. (1998). "I.J. Bienaymé : Criticality, Inequality, and Internationalization".
4985:
in nĂn variables. And there are n more linear equations from the fact that Q is a right
1114:
However, it is possible to model this scenario as a Markov process. Instead of defining
15826:
15791:
15706:
15676:
15507:
15446:
15441:
15264:
15101:
14766:
14704:
14643:
13803:
13758:
13612:
13491:
13286:
13253:
13188:
12948:
12915:
12891:
12856:
12718:
12687:
12675:
12584:
12559:
12511:
12036:
11436:
11401:
10357:
10321:
10224:
10208:
10037:
10017:
9997:
9800:
9674:
Similarly, it has been suggested that the crystallization and growth of some epitaxial
9100:
8121:
7867:
7781:
7548:
7473:
7401:
7338:
7291:
7224:
7136:
7130:
is ergodic if it is recurrent, has a period of 1, and has finite mean recurrence time.
5168:
5107:
4821:. Multiplying together stochastic matrices always yields another stochastic matrix, so
497:
446:
387:
371:
243:
132:
72:
49:
14579:
Markov Chains chapter in American Mathematical Society's introductory probability book
14433:
14362:
14325:
13649:
13373:
Handbook of Research on Modern Cryptographic Solutions for Computer and Cyber Security
13138:
12364:
12312:
10853:
Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Second Edition
10356:. In a first-order chain, the states of the system become note or pitch values, and a
8118:, a measure-preserving dynamical system is called "ergodic" iff any measurable subset
4119:
of π with a vector whose components are all 1 is unity and that π lies on a
726:
15846:
15059:
14976:
14945:
14838:
14818:
14808:
14664:
14659:
14541:
14526:
14518:, vol. 36, no. 4, pp. 52â57, ACM SIGMETRICS Performance Evaluation Review, 2009.
14504:
14496:
14487:
14473:
14463:
14410:
14377:
14348:
14305:
14290:
14272:
14037:
14030:
13882:
13858:
13808:
13377:
13291:
13208:
12992:
12953:
12896:
12837:
12793:
12758:
12723:
12655:
12622:
12589:
12486:
12428:
12408:
12391:
12370:
12318:
12260:
12179:
12148:
12123:
12104:
12085:
11947:
11911:
11875:
11831:
11804:
11777:
11744:
11714:
11656:
11594:
11567:
11471:
11367:
11334:
11291:
11229:
11197:
11148:
11127:
11106:
11086:
11062:
11035:
11008:
10981:
10954:
10926:
10916:
10878:
10857:
10830:
10803:
10659:
10309:
10248:
10240:
9836:
9817:
9385:
5084:
4986:
4826:
3206:
775:
732:
709:
686:
632:
458:
454:
414:
302:
208:
107:
15651:
15302:
12449:
11773:
11193:
7699:
has positive diagonal entries, which by previous proposition means its exponent is
5407:
row vector that represents a valid probability distribution; since the eigenvectors
4849:
unknowns, so it is computationally easier if on the one hand one selects one row in
697:
15866:
15753:
15636:
15512:
15249:
15006:
14981:
14930:
14781:
14734:
14369:
14340:
14253:
14206:
14132:
14099:
13835:
13798:
13788:
13682:
13645:
13604:
13550:
13545:
13537:
13483:
13431:
13281:
13273:
13235:
13200:
13169:
13134:
13099:
13062:
13023:
12988:
12943:
12935:
12886:
12876:
12829:
12785:
12750:
12713:
12705:
12647:
12614:
12579:
12571:
12478:
12403:
12138:
12118:
12077:
12028:
11939:
11903:
11769:
11686:
11631:
11540:
11506:
11463:
11428:
11393:
11363:
11326:
11263:
10679:
10655:
10228:
9821:
9813:
9809:
9758:
9715:, where Markov chains are in particular a central tool in the theoretical study of
9437:
9432:. A Markov matrix that is compatible with the adjacency matrix can then provide a
9417:
9380:
9374:
8689:
5103:
4120:
3649:
127:
57:
14858:
14210:
13173:
13103:
13067:
13042:
13027:
12287:
15831:
15731:
15716:
15477:
15411:
15089:
15033:
15016:
14761:
14585:
14455:
14336:
13928:
Proceedings of the 22nd International Joint Conference on Artificial Intelligence
13923:
13344:
12881:
12608:
12482:
12470:
11907:
11895:
11869:
11852:
11825:
11798:
11738:
11588:
11561:
11467:
11285:
11223:
11056:
11029:
11002:
10975:
10948:
10851:
10824:
10797:
10747:
10737:
10732:
10654:
given a sample document. Markov processes are used in a variety of recreational "
10631:
10614:
structure, rather than the 'aimless wandering' produced by a first-order system.
10325:
10232:
9896:
9869:
9863:
9728:
9445:
8987:
8979:
4789:
4329:
does not exist while the stationary distribution does, as shown by this example:
3857:
3715:
3202:
1545:
1521:
788:
784:
645:
501:
395:
203:
154:
15646:
14878:
13239:
12651:
12519:
9725:, where Markov chains have been used, e.g., to simulate the mammalian neocortex.
9640:
independent of each other, the number of molecules in state A or B at a time is
9402:
is isomorphic to a Bernoulli scheme; the Markov chain is just one such example.
4158:
converges to a rank-one matrix in which each row is the stationary distribution
15836:
15801:
15721:
15327:
15074:
14991:
14960:
14955:
14935:
14925:
14868:
14843:
14823:
14788:
14756:
14739:
14451:
14104:
14087:
13204:
12081:
10663:
10611:
10602:
A second-order Markov chain can be introduced by considering the current state
10349:
10273:
9840:
9790:
9747:
9694:
9668:
9485:
9441:
8464:
8115:
7897:
If a Markov chain has a stationary distribution, then it can be converted to a
7211:
4769:{\displaystyle \mathbf {Q} (\mathbf {P} -\mathbf {I} _{n})=\mathbf {0} _{n,n},}
2672:
which has the 'classical' Markov property by taking as state space the ordered
794:
A famous Markov chain is the so-called "drunkard's walk", a random walk on the
780:
749:
737:
713:
705:
673:
586:
505:
218:
14863:
14590:
14434:
Kemeny, John G.; Hazleton Mirkil; J. Laurie Snell; Gerald L. Thompson (1959).
14373:
14344:
14257:
13369:
13357:
12833:
12618:
12613:. Lecture Notes in Physics. Vol. 788. Springer-Verlag Berlin Heidelberg.
12575:
11943:
11330:
10320:
Markov chains can be used to model many games of chance. The children's games
9686:
Markov chains are used in various areas of biology. Notable examples include:
15899:
15738:
15279:
15111:
15069:
15011:
14833:
14749:
14689:
13686:
13212:
12283:
12227:
11545:
11528:
11381:
10635:
10353:
10305:
10304:
mobilization, etc., will generate a higher probability of transitioning from
9923:
9851:
9690:
8991:
2839:
are non-negative and describe the rate of the process transitions from state
1850:
753:
682:
664:
616:
485:
407:
91:
13839:
13793:
13509:
Simon, Herbert; C Bonini (1958). "The size distribution of business firms".
13117:
Morf, H. (1998). "The stochastic two-state solar irradiance model (STSIM)".
10930:
9879:
Numerous queueing models use continuous-time Markov chains. For example, an
8385:
An example of a non-Markovian process with a Markovian representation is an
15796:
15758:
15312:
15244:
15133:
15128:
14940:
14873:
14848:
13812:
13631:
13295:
12957:
12900:
12841:
12797:
12762:
12727:
12593:
12089:
11690:
11510:
10297:
9722:
9675:
9457:
9421:
9317:
8733:
6596:
701:
605:
567:
318:
112:
13945:
11187:
11185:
10913:
Stochastic differential equations : an introduction with applications
9001:
To find the stationary probability distribution vector, we must next find
8692:
this process has the same stationary distribution as the forward process.
4126:
15841:
15376:
15360:
15355:
15350:
15340:
15143:
15084:
15079:
15043:
14803:
14694:
13967:
12676:"Correlation analysis of enzymatic reaction of a single protein molecule"
12199:
11455:
10283:
9880:
9762:
9496:
9277:
8737:. Each element of the one-step transition probability matrix of the EMC,
8389:
7355:
is zero or positive, and therefore can be found on a directed graph with
5114:
and proceed with a bit more involved set of arguments in a similar way.)
4810:
4116:
3911:
795:
771:
612:
537:
520:
390:
sequence, in which the chain moves state at discrete time steps, gives a
379:
238:
79:
67:
11563:
Paul Lévy and Maurice Fréchet: 50 Years of Correspondence in 107 Letters
11559:
11267:
4228:
is the column vector with all entries equal to 1. This is stated by the
15851:
15391:
15335:
15219:
15172:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
14612:
14282:
14230:
Izvestiya Fiziko-matematicheskogo obschestva pri Kazanskom universitete
13616:
13541:
13495:
13449:
Page, Lawrence; Brin, Sergey; Motwani, Rajeev; Winograd, Terry (1999).
12740:
12709:
12040:
12016:
11440:
11405:
10293:
9702:
7207:
4026:
3727:
96:
42:
14062:"Forecasting oil price trends using wavelets and hidden Markov models"
14060:
de Souza e Silva, E.G.; Legey, L.F.L.; de Souza e Silva, E.A. (2010).
13435:
13277:
12939:
12789:
12754:
11635:
10207:(MCMC). In recent years this has revolutionized the practicability of
9808:
system perfectly, such signal models can make possible very effective
9303:(2ÎŽ), ... give the sequence of states visited by the ÎŽ-skeleton.
6859:, there is a non-zero probability that the chain will never return to
5070:{\displaystyle {\boldsymbol {\pi }}={\boldsymbol {\pi }}\mathbf {P} ,}
519:
A Markov chain is a type of Markov process that has either a discrete
15345:
14402:
in 1963 and translated to English with the assistance of the author.)
14399:
14136:
14059:
13756:
11529:"Half a Century with Probability Theory: Some Personal Recollections"
10639:
10301:
10277:
9883:
is a CTMC on the non-negative integers where upward transitions from
9663:
9655:
4214:{\displaystyle \lim _{k\to \infty }\mathbf {P} ^{k}=\mathbf {1} \pi }
1978:{\displaystyle \Pr(X_{n+1}=x\mid X_{n}=y)=\Pr(X_{n}=x\mid X_{n-1}=y)}
438:
434:
13608:
13487:
13189:"A Systematic Review of Hidden Markov Models and Their Applications"
12032:
11800:
The Wonderful world of stochastics: a tribute to Elliott W. Montroll
11432:
11397:
9868:
Markov chains are the basis for the analytical treatment of queues (
9731:, for instance with the modeling of viral infection of single cells.
4131:
If the Markov chain is time-homogeneous, then the transition matrix
472:
are used to describe something that is related to a Markov process.
10606:
also the previous state, as indicated in the second table. Higher,
10341:
10227:
built a Markov chain model of the distribution of income in 1953.
9987:
7126:
if it is aperiodic and positive recurrent. In other words, a state
6643:
if the probability of leaving the class is zero. A Markov chain is
6576:
can also speed up this convergence to the stationary distribution.
5034:
3164:{\displaystyle \Pr(X(t+h)=j\mid X(t)=i)=\delta _{ij}+q_{ij}h+o(h),}
2902:
be the random variable describing the state of the process at time
641:
512:
on the present state of the system, its future and past states are
375:
223:
14088:"Markov chain modeling for very-short-term wind power forecasting"
13902:
13370:
Gupta, Brij; Agrawal, Dharma P.; Yamaguchi, Shingo (16 May 2016).
12692:
12558:
van Ravenzwaaij, Don; Cassey, Pete; Brown, Scott D. (2016-03-11).
10822:
9994:
is defined by a Markov chain. It is the probability to be at page
8378:
has the Markov property, then it is a Markovian representation of
4876:
with its right-most column replaced with all 1's. If exists then
3730:, the transition probability distribution can be represented by a
2773:{\displaystyle Y_{n}=\left(X_{n},X_{n-1},\ldots ,X_{n-m+1}\right)}
560:(discrete-time) Markov chain on a countable or finite state space
13251:
12366:
Non-negative matrices; an introduction to theory and applications
12314:
Non-negative matrices; an introduction to theory and applications
11992:
Proceedings of the 14th Symposium on Reliable Distributed Systems
11867:
8720:
6981:{\displaystyle M_{i}=E=\sum _{n=1}^{\infty }n\cdot f_{ii}^{(n)}.}
450:
442:
430:
11652:
Continuous-Time Markov Chains: An Applications-Oriented Approach
9850:
lossless data compression algorithm combines Markov chains with
600:, but a few authors use the term "Markov process" to refer to a
30:
12450:"Smoothing of noisy AR signals using an adaptive Kalman filter"
11560:
Marc Barbut; Bernard Locker; Laurent Mazliak (23 August 2016).
11225:
Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues
10910:
10345:
10129:{\displaystyle {\frac {\alpha }{k_{i}}}+{\frac {1-\alpha }{N}}}
9991:
9975:{\displaystyle {\frac {\alpha }{k_{i}}}+{\frac {1-\alpha }{N}}}
9706:
8964:{\displaystyle S=I-\left(\operatorname {diag} (Q)\right)^{-1}Q}
4646:{\textstyle \mathbf {Q} =\lim _{k\to \infty }\mathbf {P} ^{k}.}
840:
represents the total value of the coins set on the table after
403:
13709:
Department of Finance, the Anderson School of Management, UCLA
13041:
Bright, J. M.; Smith, C. I.; Taylor, P. G.; Crook, R. (2015).
12855:
George, Dileep; Hawkins, Jeff (2009). Friston, Karl J. (ed.).
12144:
Stochastic Cellular Systems: Ergodicity, Memory, Morphogenesis
11491:
11489:
11487:
10849:
9678:
oxide materials can be accurately described by Markov chains.
9440:
are isomorphic to topological Markov chains; examples include
6774:{\displaystyle k=\gcd\{n>0:\Pr(X_{n}=i\mid X_{0}=i)>0\}}
6549:{\displaystyle |\lambda _{2}|\geq \cdots \geq |\lambda _{n}|,}
5363:
which solves the stationary distribution equation above). Let
3858:
Stationary distribution relation to eigenvectors and simplices
14316:. Second edition to appear, Cambridge University Press, 2009.
14268:. Original edition published by Addison-Wesley; reprinted by
14085:
12810:
12363:
Seneta, E. (Eugene) (1973). "2.4. Combinatorial properties".
11736:
11675:
11495:
11181:
11179:
11177:
11175:
11173:
11000:
10366:
8236:
iff its corresponding measure-preserving dynamical system is
7470:. The only case where it is an equality is when the graph of
12560:"A simple introduction to Markov Chain MonteâCarlo sampling"
8699:
if the reversed process is the same as the forward process.
4864:
Here is one method for doing so: first, define the function
3289:, ... to describe holding times in each of the states where
14188:
13825:
13252:
Pratas, D; Silva, R; Pinho, A; Ferreira, P (May 18, 2015).
12970:
11669:
11484:
11384:(1996). "Markov and the Birth of Chain Dependence Theory".
11347:
10795:
10362:
8880:
480:
15152:
Autoregressive conditional heteroskedasticity (ARCH) model
14086:
Carpinone, A; Giorgio, M; Langella, R.; Testa, A. (2015).
13474:
Champernowne, D (1953). "A model of income distribution".
12857:"Towards a Mathematical Theory of Cortical Micro-circuits"
12557:
11844:
11732:
11730:
11170:
10823:
Reuven Y. Rubinstein; Dirk P. Kroese (20 September 2011).
10816:
9789:
Markov chains are used throughout information processing.
8069:. Similarly we can construct such a dynamical system with
5077:(if exists) the stationary (or steady state) distribution
1830:{\displaystyle \Pr(X_{1}=x_{1},\ldots ,X_{n}=x_{n})>0.}
14417:. Appendix contains abridged Meyn & Tweedie. online:
13757:
Acemoglu, Daron; Georgy Egorov; Konstantin Sonin (2011).
11279:
11277:
11031:
Stochastic processes: a survey of the mathematical theory
9644:
times the probability a given molecule is in that state.
7535:{\displaystyle 1\to 2\to \dots \to n\to 1{\text{ and }}2}
4981:
Explain: The original matrix equation is equivalent to a
3239:
can be seen as measuring how quickly the transition from
14680:
Independent and identically distributed random variables
13879:
Formalized Music: Mathematics and Thought in Composition
13451:
The PageRank Citation Ranking: Bringing Order to the Web
12775:
12457:
9th European Signal Processing Conference (EUSIPCO 1998)
11254:
Hayes, Brian (2013). "First links in the Markov chain".
10215:
to be simulated and their parameters found numerically.
9291:) at intervals of ÎŽ units of time. The random variables
5027:
will have the 1 and the 0's in the same positions as in
4972:{\displaystyle \mathbf {Q} =f(\mathbf {0} _{n,n})^{-1}.}
3336:= 0, 1, 2, 3, ... and times indexed up to this value of
13566:
Fama, E (1965). "The behavior of stock market prices".
13302:
11868:
Donald L. Snyder; Michael I. Miller (6 December 2012).
11727:
11702:
11700:
11590:
Basic Principles and Applications of Probability Theory
5215:{\displaystyle \mathbf {P} =\mathbf {U\Sigma U} ^{-1}.}
4127:
Time-homogeneous Markov chain with a finite state space
2859:
There are three equivalent definitions of the process.
2246:
A Markov chain with memory (or a Markov chain of order
15157:
Autoregressive integrated moving average (ARIMA) model
14237:
Dynamic Probabilistic Systems, volume 1: Markov Chains
13528:
Bachelier, Louis (1900). "Théorie de la spéculation".
13187:
Mor, Bhavya; Garhwal, Sunita; Kumar, Ajay (May 2021).
13040:
11655:. Springer Science & Business Media. p. vii.
11593:. Springer Science & Business Media. p. 146.
11553:
11353:
11274:
11048:
10994:
10791:
10789:
9551:
7892:
6647:
if there is one communicating class, the state space.
4837:. Including the fact that the sum of each the rows in
4603:
4535:
4496:
4458:
4352:
4290:
4238:
4072:
4040:
3980:
1989:. The probability of the transition is independent of
1176:
of the various coin types on the table. For instance,
578:
Continuous-time Markov process or Markov jump process
421:, which are used for simulating sampling from complex
11874:. Springer Science & Business Media. p. 32.
11796:
11580:
11228:. Springer Science & Business Media. p. ix.
11217:
11215:
11213:
11186:
Charles Miller Grinstead; James Laurie Snell (1997).
10300:, the ratio of urban to rural residence, the rate of
10177:
10142:
10087:
10060:
10040:
10020:
10000:
9933:
9514:
9147:
9106:
9085:
9065:
9030:
9007:
8910:
8771:
8639:
8484:
8437:
8273:
8215:
8189:
8144:
8124:
8075:
7995:
7963:
7942:
7907:
7870:
7808:
7784:
7740:
7705:
7678:
7638:
7597:
7571:
7551:
7496:
7476:
7428:
7404:
7361:
7341:
7314:
7294:
7247:
7227:
7159:
7139:
7059:
7042:
if there are no outgoing transitions from the state.
7005:
6888:
6820:
6794:
6693:
6493:
5614:
5457:
5422:
5234:
5179:
5045:
4885:
4710:
4664:
4452:
4338:
4171:
4039:
3923:
3879:
3759:
3665:
3398:
3250:
3215:
3177:
3053:
2992:
2953:
2920:
2881:
2686:
2645:
2612:
2602:
In other words, the future state depends on the past
2260:
2222:
2002:
1871:
1756:
1557:
1456:
1411:
1365:
1338:
1311:
1266:
1227:
1182:
1151:
1120:
1090:
1063:
1036:
1000:
973:
937:
883:
850:
819:
14536:
G. Bolch, S. Greiner, H. de Meer and K. S. Trivedi,
14009:(November 1984). "A Travesty Generator for Micros".
12137:
11790:
11697:
11642:
10953:. Springer Science & Business Media. p. 7.
10942:
10940:
10171:
for all pages that are not linked to. The parameter
8364:{\displaystyle Y(t)={\big \{}X(s):s\in \,{\big \}}.}
8062:{\displaystyle T(X_{0},X_{1},\dots )=(X_{1},\dots )}
6622:(probabilistic cellular automata). See for instance
6605:
4581:(This example illustrates a periodic Markov chain.)
4139:-step transition probability can be computed as the
3327:
744:
derived in a 1928 paper an equation, now called the
13930:, IJCAI, pages 635â642, Barcelona, Spain, July 2011
13530:
Annales Scientifiques de l'Ăcole Normale SupĂ©rieure
13005:
12204:"Show that positive recurrence is a class property"
11737:Samuel Karlin; Howard E. Taylor (2 December 2012).
11648:
11001:Samuel Karlin; Howard E. Taylor (2 December 2012).
10876:
10786:
8471:that the chain enters one of the states in the set
1865:Time-homogeneous Markov chains are processes where
402:(CTMC). Markov processes are named in honor of the
14435:
14361:
14324:
14029:
14004:
13662:
13390:
12392:"An improvement of the Dulmage-Mendelsohn theorem"
12176:Stochastic Models in Operations Research, Volume 1
12059:
11458:, Seneta E, Crépel P, Fienberg SE, Gani J (eds.).
11304:
11210:
11075:
10877:
10183:
10163:
10128:
10073:
10046:
10026:
10006:
9974:
9899:and describe job arrivals, while transitions from
9701:use continuous-time Markov chains to describe the
9616:
9253:
9125:
9091:
9071:
9048:
9013:
8963:
8886:
8762:. These conditional probabilities may be found by
8680:
8608:
8455:
8363:
8252:be a non-Markovian process. Then define a process
8221:
8201:
8175:
8130:
8096:
8061:
7981:
7948:
7928:
7876:
7856:
7790:
7770:
7723:
7691:
7664:
7621:
7583:
7557:
7534:
7482:
7462:
7410:
7387:
7347:
7327:
7300:
7260:
7233:
7172:
7145:
7102:
7018:
6980:
6832:
6806:
6773:
6548:
6414:
5545:
5440:
5341:
5214:
5069:
4971:
4768:
4686:
4645:
4570:
4437:
4321:
4269:
4213:
4107:
4054:
4009:
3964:{\displaystyle \pi ={\frac {e}{\sum _{i}{e_{i}}}}}
3963:
3914:we see that the two concepts are related and that
3899:
3832:{\displaystyle p_{ij}=\Pr(X_{n+1}=j\mid X_{n}=i).}
3831:
3703:
3628:
3231:
3193:
3163:
3027:
2978:
2939:
2894:
2772:
2664:
2631:
2594:
2235:
2200:
1977:
1829:
1738:
1493:
1442:
1396:
1351:
1324:
1297:
1251:
1213:
1164:
1133:
1103:
1076:
1049:
1022:
986:
959:
916:
869:
832:
611:While the time parameter is usually discrete, the
14388:(NB. This was originally published in Russian as
13917:"Finite-Length Markov Processes with Constraints"
13730:"A Markov Chain Example in Credit Risk Modelling"
13308:
11586:
11163:Meyn, S. Sean P., and Richard L. Tweedie. (2009)
11021:
10967:
10937:
9559:
9558:
9541:
9540:
9306:
4322:{\textstyle \lim _{k\to \infty }\mathbf {P} ^{k}}
4270:{\textstyle \lim _{k\to \infty }\mathbf {P} ^{k}}
2243:is a stationary distribution of the Markov chain.
1084:are impacted by our knowledge of values prior to
15897:
15039:Stochastic chains with memory of variable length
13193:Archives of Computational Methods in Engineering
12607:Gattringer, Christof; Lang, Christian B (2010).
12512:Ergodic Theory: Basic Examples and Constructions
10850:Dani Gamerman; Hedibert F. Lopes (10 May 2006).
10843:
6718:
6700:
5035:Convergence speed to the stationary distribution
5011:on its main diagonal that is equal to 1 and the
4613:
4292:
4240:
4173:
3776:
3399:
3054:
2811:is defined by a finite or countable state space
2422:
2271:
2096:
2003:
1925:
1872:
1757:
1676:
1558:
1450:state depends exclusively on the outcome of the
712:in 1907, and a branching process, introduced by
425:, and have found application in areas including
14166:Cambridge: National Bureau of Economic Research
13763:Proceedings of the National Academy of Sciences
13702:"Stock Price Volatility and the Equity Premium"
13186:
13151:
13081:
12914:Gupta, Ankur; Rawlings, James B. (April 2014).
11989:
11757:
11713:. John Wiley & Sons. pp. 373 and 374.
11283:
11167:. Cambridge University Press. (Preface, p. iii)
11054:
10272:Markov chains are generally used in describing
10081:links to it then it has transition probability
9828:. Markov chains also play an important role in
9753:
3846:sums to one and all elements are non-negative,
2784:
676:who claimed independence was necessary for the
14521:R. A. Sahner, K. S. Trivedi and A. Puliafito,
14503:, John Wiley & Sons, Inc. New York, 2002.
14270:Society for Industrial and Applied Mathematics
13396:
12606:
12516:Encyclopedia of Complexity and Systems Science
11850:
11823:
11706:
11607:
11522:
11520:
11221:
10973:
10946:
10870:
10796:Sean Meyn; Richard L. Tweedie (2 April 2009).
10621:limitation, a new approach has been proposed.
9839:for error correction), speech recognition and
9799:, which in a single step created the field of
9079:being a row vector, such that all elements in
8097:{\displaystyle \Omega =\Sigma ^{\mathbb {Z} }}
7929:{\displaystyle \Omega =\Sigma ^{\mathbb {N} }}
5129:be the diagonal matrix of left eigenvalues of
1520:A discrete-time Markov chain is a sequence of
14628:
14389:
13942:"MARKOV CHAIN MODELS: THEORETICAL BACKGROUND"
13630:Calvet, Laurent E.; Fisher, Adlai J. (2001).
13508:
12913:
12854:
12641:
12520:https://doi.org/10.1007/978-0-387-30440-3_177
11454:Bru B, Hertz S (2001). "Maurice Fréchet". In
11447:
11310:
11249:
11247:
11245:
11027:
8633:, the time-reversed process is defined to be
8353:
8291:
7734:(Dulmage-Mendelsohn theorem) The exponent is
5225:Let the eigenvalues be enumerated such that:
3910:By comparing this definition with that of an
3361:, ... and all states recorded at these times
1996:Stationary Markov chains are processes where
1509:
343:
14032:Virtual Muse: Experiments in Computer Poetry
13852:
13629:
13473:
13228:Studies in History and Philosophy of Science
12778:Journal of Chemical Information and Modeling
12743:Journal of Chemical Information and Modeling
12644:Design and Analysis of Biomolecular Circuits
12473:(1997). "Continuous-time Markov chains II".
12174:Heyman, Daniel P.; Sobel, Mathew J. (1982).
11830:. Courier Dover Publications. p. 7, 8.
11462:. New York, NY: Springer. pp. 331â334.
10669:
10281:
10034:is the number of known webpages, and a page
9361:Partially observable Markov decision process
9114:
9107:
8243:
6768:
6703:
2606:states. It is possible to construct a chain
911:
884:
12173:
11898:(1997). "Continuous-time Markov chains I".
11817:
11679:Bulletin of the London Mathematical Society
11517:
11499:Bulletin of the London Mathematical Society
11412:
11374:
10980:. Courier Dover Publications. p. 188.
9737:for disease outbreak and epidemic modeling.
4687:{\displaystyle \mathbf {QP} =\mathbf {Q} .}
2862:
1145:of the coins on the table, we could define
917:{\displaystyle \{X_{n}:n\in \mathbb {N} \}}
585:with the Markov property (for example, the
15167:Autoregressiveâmoving-average (ARMA) model
14635:
14621:
14036:. Hanover, NH: Wesleyan University Press.
13397:Langville, Amy N.; Meyer, Carl D. (2006).
13363:
12533:"Thermodynamics and Statistical Mechanics"
12422:
12259:. San Francisco: Holden-Day. p. 145.
11242:
11145:The Oxford Dictionary of Statistical Terms
11124:The Oxford Dictionary of Statistical Terms
10638:and differences when playing on grass vs.
9416:When the Markov matrix is replaced by the
9405:
8475:) is the minimal non-negative solution to
4701:from both sides and factoring then yields
3734:, called the transition matrix, with the (
350:
336:
27:Random process independent of past history
14252:(4). Translated by Link, David: 591â600.
14103:
13998:
13915:Pachet, F.; Roy, P.; Barbieri, G. (2011)
13802:
13792:
13782:
13676:
13598:
13549:
13527:
13458:
13425:
13285:
13066:
12947:
12890:
12880:
12717:
12691:
12583:
12510:Matthew Nicol and Karl Petersen, (2009) "
12447:
12407:
12122:
12014:
11999:
11764:Weiss, George H. (2006). "Random Walks".
11613:
11544:
11320:
10802:. Cambridge University Press. p. 3.
10645:
9854:to achieve very high compression ratios.
8414:
8350:
8088:
7920:
6586:Markov chains on a measurable state space
5536:
5425:
3028:{\displaystyle \left(X_{s}:s<t\right)}
907:
531:
14642:
14232:, 2-ya seriya, tom 15, pp. 135â156.
14150:
14121:"Quantitative Terrorism Risk Assessment"
14112:
14081:
14079:
14021:
13584:
13442:
13245:
12553:
12551:
12506:
12504:
12502:
11871:Random Point Processes in Time and Space
11453:
10911:Ăksendal, B. K. (Bernt Karsten) (2003).
10652:generate superficially real-looking text
10218:
9922:
9918:
9647:The classical model of enzyme activity,
9484:Markovian systems appear extensively in
8998:and setting all other elements to zero.
8706:
8260:represents a time-interval of states of
4033:The values of a stationary distribution
2866:
564:Markov chain on a measurable state space
479:
413:Markov chains have many applications as
29:
14484:Non-negative matrices and Markov chains
14426:Sequential Machines and Automata Theory
14407:Control Techniques for Complex Networks
14184:
14182:
14055:
14053:
14027:
13877:Xenakis, Iannis; Kanach, Sharon (1992)
13699:
13399:"A Reordering for the PageRank Problem"
13338:Control Techniques for Complex Networks
13320:MacTutor History of Mathematics Archive
13225:
12674:Du, Chao; Kou, S. C. (September 2012).
12530:
12463:
12282:
12102:
11933:
11929:
11927:
11888:
11649:William J. Anderson (6 December 2012).
11614:Bernstein, Jeremy (2005). "Bachelier".
9843:(such as in rearrangements detection).
9761:variability assessments are useful for
9534:
8264:. Mathematically, this takes the form:
7591:diagonal entries, then its exponent is
7288:, of a regular matrix, is the smallest
7275:
5621:
5094:is diagonalizable or equivalently that
5055:
5047:
4108:{\textstyle \sum _{i}1\cdot \pi _{i}=1}
4062:are associated with the state space of
2906:, and assume the process is in a state
807:
14:
15898:
15473:Doob's martingale convergence theorems
14360:
14319:
14302:Markov Chains and Stochastic Stability
14243:
12469:
12362:
12310:
12254:
12226:
12178:. New York: McGraw-Hill. p. 230.
12167:
12141:; Kryukov, V. I.; Toom, A. L. (1978).
11936:Basics of Applied Stochastic Processes
11894:
11740:A First Course in Stochastic Processes
11526:
11418:
11380:
11192:. American Mathematical Soc. pp.
11165:Markov chains and stochastic stability
11083:The Cambridge Dictionary of Statistics
11007:. Academic Press. pp. 29 and 30.
11004:A First Course in Stochastic Processes
10829:. John Wiley & Sons. p. 225.
10799:Markov Chains and Stochastic Stability
10164:{\displaystyle {\frac {1-\alpha }{N}}}
9803:, opens by introducing the concept of
9796:A Mathematical Theory of Communication
9662:Also, the growth (and composition) of
8681:{\displaystyle {\hat {X}}_{t}=X_{T-t}}
6672:of the number of transitions by which
6579:
6397:
6320:
6249:
6178:
6079:
6026:
5979:
5890:
5852:
5820:
5466:
3900:{\displaystyle \pi \mathbf {P} =\pi .}
15225:Constant elasticity of variance (CEV)
15215:ChanâKarolyiâLongstaffâSanders (CKLS)
14616:
14602:A visual explanation of Markov Chains
14423:
14076:
13846:
13759:"Political model of social evolution"
13632:"Forecasting Multifractal Volatility"
12673:
12610:Quantum Chromodynamics on the Lattice
12548:
12499:
12369:. Internet Archive. New York, Wiley.
12317:. Internet Archive. New York, Wiley.
12278:
12276:
12017:"Convergence Rates for Markov Chains"
11763:
11587:Valeriy Skorokhod (5 December 2005).
11253:
11034:. Springer-Verlag. pp. 106â121.
10826:Simulation and the Monte Carlo Method
10650:Markov processes can also be used to
10136:for all pages that are linked to and
9784:
9769:
9334:System state is partially observable
9276:is found, it must be normalized to a
5039:As stated earlier, from the equation
3721:
3268:th jump of the process and variables
1856:called the state space of the chain.
1252:{\displaystyle 6\times 6\times 6=216}
14525:, Kluwer Academic Publishers, 1996.
14409:. Cambridge University Press, 2007.
14300:S. P. Meyn and R. L. Tweedie (1993)
14179:
14050:
13565:
13406:SIAM Journal on Scientific Computing
13330:
13116:
12646:, Springer New York, pp. 3â42,
12389:
11924:
11766:Encyclopedia of Statistical Sciences
11710:Probability and Stochastic Processes
10898:participating institution membership
7982:{\displaystyle T:\Omega \to \Omega }
6487:exponentially. This follows because
4833:is a stochastic matrix to solve for
4143:-th power of the transition matrix,
4135:is the same after each step, so the
3255:Define a discrete-time Markov chain
1504:
752:, starting in 1930s, and then later
650:conditional probability distribution
14538:Queueing Networks and Markov Chains
14118:
13347:, Cambridge University Press, 2007.
9368:
9049:{\displaystyle \varphi S=\varphi ,}
8713:stationary probability distribution
7899:measure-preserving dynamical system
7893:Measure-preserving dynamical system
7857:{\displaystyle \leq (d+1)+s(d+1-2)}
7268:are positive. Some authors call it
6871:) otherwise. For a recurrent state
4055:{\displaystyle \textstyle \pi _{i}}
3714:with initial condition P(0) is the
2254:is finite, is a process satisfying
602:continuous-time Markov chain (CTMC)
24:
15712:Skorokhod's representation theorem
15493:Law of large numbers (weak/strong)
13965:
13939:
12273:
11707:Ionut Florescu (7 November 2014).
10915:(6th ed.). Berlin: Springer.
10369:), or any other desirable metric.
10267:
10211:methods, allowing a wide range of
9857:
9597:
9574:
9424:, the resulting shift is termed a
8216:
8196:
8083:
8076:
7976:
7970:
7943:
7915:
7908:
7665:{\displaystyle \mathrm {sign} (M)}
7649:
7646:
7643:
7640:
7388:{\displaystyle \mathrm {sign} (M)}
7372:
7369:
7366:
7363:
6940:
6098:
4623:
4302:
4250:
4183:
3251:Jump chain/holding time definition
2986:is independent of previous values
1014:
951:
552:Continuous or general state space
25:
15932:
15682:Martingale representation theorem
14553:
14540:, John Wiley, 2nd edition, 2006.
13665:Journal of Financial Econometrics
12564:Psychonomic Bulletin & Review
12425:Foundations of modern probability
12198:
12105:"Interaction of Markov Processes"
10904:
10703:Markov chain approximation method
9911: > 1) occur at rate
9331:System state is fully observable
7045:
6612:locally interacting Markov chains
6606:Locally interacting Markov chains
6473:â â with a speed in the order of
5441:{\displaystyle \mathbb {R} ^{n},}
4017:) multiple of a left eigenvector
3654:first-order differential equation
3328:Transition probability definition
1494:{\displaystyle X_{n-1}=\ell ,m,p}
1023:{\displaystyle X_{7}\geq \$ 0.60}
598:discrete-time Markov chain (DTMC)
15727:Stochastic differential equation
15617:Doob's optional stopping theorem
15612:DoobâMeyer decomposition theorem
14289:. New York: John Wiley and Sons
13980:
13959:
13933:
13700:Brennan, Michael; Xiab, Yihong.
12814:Acta Crystallographica Section A
12680:The Annals of Applied Statistics
12448:Doblinger, G. (September 1998).
11803:. North-Holland. pp. 8â10.
11421:International Statistical Review
11386:International Statistical Review
11368:10.1111/j.1751-5823.2012.00181.x
11356:International Statistical Review
9126:{\displaystyle \|\varphi \|_{1}}
8619:
8431:of hitting times (where element
8395:
7463:{\displaystyle \leq (n-1)^{2}+1}
6590:
6386:
6309:
6238:
6167:
6068:
6015:
5968:
5918:
5906:
5903:
5879:
5841:
5809:
5769:
5757:
5754:
5751:
5722:
5719:
5716:
5690:
5687:
5684:
5661:
5658:
5655:
5645:
5508:
5460:
5196:
5193:
5190:
5181:
5060:
4940:
4931:
4902:
4887:
4747:
4729:
4720:
4712:
4677:
4669:
4666:
4630:
4605:
4431:
4408:
4386:
4340:
4309:
4257:
4204:
4190:
4010:{\textstyle \sum _{i}\pi _{i}=1}
3884:
2795:A continuous-time Markov chain (
378:of possible events in which the
56:
15597:Convergence of random variables
15483:FisherâTippettâGnedenko theorem
14516:SHARPE at the age of twenty-two
14242:Classical Text in Translation:
14092:Electric Power Systems Research
13909:
13891:
13871:
13819:
13750:
13722:
13693:
13656:
13623:
13578:
13558:
13521:
13502:
13467:
13350:
13219:
13180:
13145:
13110:
13075:
13034:
12999:
12964:
12907:
12848:
12804:
12769:
12734:
12667:
12635:
12600:
12524:
12441:
12416:
12383:
12356:
12331:
12304:
12248:
12220:
12192:
12147:. Manchester University Press.
12131:
12096:
12053:
12008:
11983:
11960:
11861:
11824:Emanuel Parzen (17 June 2015).
11774:10.1002/0471667196.ess2180.pub2
11290:. Wiley. pp. 235 and 358.
11222:Pierre Bremaud (9 March 2013).
11157:
11137:
11116:
11095:
11061:. Wiley. pp. 174 and 231.
10974:Emanuel Parzen (17 June 2015).
10693:Dynamics of Markovian particles
9705:present at a given site in the
9471:
9311:
7901:: Let the probability space be
6624:Interaction of Markov Processes
5521:
4994:One thing to notice is that if
4405:
4383:
1305:. The probability of achieving
620:
15195:Binomial options pricing model
14514:K. S. Trivedi and R.A.Sahner,
14438:Finite Mathematical Structures
12015:Rosenthal, Jeffrey S. (1995).
11797:Michael F. Shlesinger (1985).
11743:. Academic Press. p. 49.
11566:. Springer London. p. 5.
11460:Statisticians of the Centuries
11313:A Festschrift for Herman Rubin
10950:Applied Probability and Queues
10947:SĂžren Asmussen (15 May 2003).
10336:Markov chains are employed in
9590:
9561:
9536:
9436:on the subshift. Many chaotic
9307:Special types of Markov chains
9236:
9223:
9219:
9213:
9204:
9197:
9182:
9178:
9172:
9163:
8941:
8935:
8723:continuous-time Markov chain,
8647:
8347:
8344:
8338:
8329:
8323:
8317:
8305:
8299:
8283:
8277:
8164:
8158:
8056:
8037:
8031:
7999:
7973:
7851:
7833:
7824:
7812:
7765:
7753:
7659:
7653:
7518:
7512:
7506:
7500:
7445:
7432:
7418:be the number of states, then
7382:
7376:
7221:iff there exists some integer
7201:
7097:
7084:
6970:
6964:
6918:
6905:
6759:
6721:
6678:can be reached, starting from
6539:
6524:
6510:
6495:
5632:
5626:
5332:
5317:
5303:
5288:
5280:
5265:
5257:
5242:
4983:system of nĂn linear equations
4954:
4950:
4927:
4921:
4918:
4897:
4739:
4716:
4620:
4299:
4247:
4180:
3823:
3779:
3695:
3689:
3680:
3674:
3623:
3591:
3552:
3402:
3155:
3149:
3105:
3096:
3090:
3075:
3063:
3057:
2659:
2646:
2626:
2613:
2571:
2425:
2408:
2274:
2195:
2099:
2090:
2006:
1972:
1928:
1919:
1875:
1818:
1760:
1750:are well defined, that is, if
1730:
1679:
1670:
1561:
626:
123:Collectively exhaustive events
13:
1:
15662:Kolmogorov continuity theorem
15498:Law of the iterated logarithm
14592:Introduction to Markov Chains
14444:Classical text. cf Chapter 6
14221:
14211:10.1016/j.solener.2019.04.014
14157:Woo, Gordon (December 2003).
13988:"Poet's Corner â Fieralingue"
13650:10.1016/S0304-4076(01)00069-0
13376:. IGI Global. pp. 448â.
13174:10.1016/j.solener.2018.05.055
13139:10.1016/S0038-092X(98)00004-8
13104:10.1016/j.solener.2018.07.056
13068:10.1016/j.solener.2015.02.032
13028:10.1016/j.solener.2014.02.026
12339:"10.3: Regular Markov Chains"
12232:"Markov Chains: Basic Theory"
10763:Stochastic cellular automaton
10338:algorithmic music composition
10245:Markov switching multifractal
10198:
9891: + 1 occur at rate
9400:stationary stochastic process
7771:{\displaystyle \leq n+s(n-2)}
7113:
6629:
5083:is a left eigenvector of row
4280:For some stochastic matrices
4154:. Additionally, in this case
1859:
960:{\displaystyle X_{6}=\$ 0.50}
583:continuous stochastic process
504:(sometimes characterized as "
491:
475:
15667:Kolmogorov extension theorem
15346:Generalized queueing network
14854:Interacting particle systems
14368:. Vol. II (122). 1965.
14159:"Insuring Against Al-Quaeda"
14156:
13968:"BASEBALL AS A MARKOV CHAIN"
12993:10.1016/0038-092X(88)90049-7
12882:10.1371/journal.pcbi.1000532
12483:10.1017/CBO9780511810633.005
12409:10.1016/0012-365X(95)00060-A
12311:Seneta, E. (Eugene) (1973).
12124:10.1016/0001-8708(70)90034-4
11968:"Chapter 11 "Markov Chains""
11908:10.1017/CBO9780511810633.004
11468:10.1007/978-1-4613-0179-0_71
9820:. They also allow effective
9779:automatic speech recognition
9754:Solar irradiance variability
9502:
9393:Ornstein isomorphism theorem
8754:of transitioning from state
8202:{\displaystyle S=\emptyset }
7103:{\displaystyle \pi _{i}=1/E}
6620:stochastic cellular automata
6438:(normalized by L2 norm) and
3194:{\displaystyle \delta _{ij}}
2791:Continuous-time Markov chain
2785:Continuous-time Markov chain
400:continuous-time Markov chain
7:
14799:Continuous-time random walk
14567:Encyclopedia of Mathematics
14125:The Journal of Risk Finance
13855:The Computer Music Tutorial
13240:10.1016/j.shpsa.2008.12.011
12652:10.1007/978-1-4419-6766-4_1
11616:American Journal of Physics
11189:Introduction to Probability
10773:Variable-order Markov model
10685:
10624:
10237:efficient-market hypothesis
10191:is taken to be about 0.15.
8729:embedded Markov chain (EMC)
8392:of order greater than one.
8176:{\displaystyle T^{-1}(S)=S}
7622:{\displaystyle \leq 2n-k-1}
7217:Some authors call a matrix
6616:interacting particle system
5389:is the left eigenvector of
3870:on it and so is defined by
3704:{\displaystyle P'(t)=P(t)Q}
1443:{\displaystyle X_{n}=i,j,k}
1397:{\displaystyle X_{2}=1,0,1}
1298:{\displaystyle X_{1}=0,1,0}
1214:{\displaystyle X_{6}=1,0,5}
759:
746:ChapmanâKolmogorov equation
525:discrete or continuous time
10:
15937:
15807:Extreme value theory (EVT)
15607:Doob decomposition theorem
14899:OrnsteinâUhlenbeck process
14670:Chinese restaurant process
14462:, D. van Nostrand Company
14304:. London: Springer-Verlag
14119:Woo, Gordon (2002-04-01).
14105:10.1016/j.epsr.2014.12.025
13853:Curtis Roads, ed. (1996).
13205:10.1007/s11831-020-09422-4
12208:Mathematics Stack Exchange
12082:10.1103/PhysRevE.84.041112
11134:(entry for "Markov chain")
10708:Markov chain geostatistics
9861:
9741:
9681:
9495:Markov chains are used in
9479:
9409:
9372:
9315:
8727:, is by first finding its
8711:One method of finding the
8399:
8256:, such that each state of
7395:as its adjacency matrix.
6583:
4845:equations for determining
3862:A stationary distribution
2788:
1516:Discrete-time Markov chain
1513:
1510:Discrete-time Markov chain
774:based on integers and the
763:
659:
392:discrete-time Markov chain
15875:
15779:
15687:Optional stopping theorem
15584:
15546:
15488:Large deviation principle
15455:
15369:
15326:
15293:
15240:HeathâJarrowâMorton (HJM)
15185:
15177:Moving-average (MA) model
15162:Autoregressive (AR) model
15142:
15052:
14987:Hidden Markov model (HMM)
14969:
14921:SchrammâLoewner evolution
14725:
14650:
14424:Booth, Taylor L. (1967).
14390:
14374:10.1007/978-3-662-25360-1
14345:10.1007/978-3-662-00031-1
14321:Dynkin, Eugene Borisovich
14258:10.1017/s0269889706001074
14028:Hartman, Charles (1996).
12834:10.1107/S0108767311044874
12619:10.1007/978-3-642-01850-3
12576:10.3758/s13423-016-1015-8
12423:Kallenberg, Olav (2002).
12390:Shen, Jian (1996-10-15).
11944:10.1007/978-3-540-89332-5
11934:Serfozo, Richard (2009).
11533:The Annals of Probability
10885:Oxford English Dictionary
10728:Markov information source
10718:Markov chain tree theorem
10670:Probabilistic forecasting
9649:MichaelisâMenten kinetics
9627:Michaelis-Menten kinetics
9450:ProuhetâThueâMorse system
9268:may be periodic, even if
8456:{\displaystyle k_{i}^{A}}
8244:Markovian representations
7724:{\displaystyle \leq 2n-2}
7308:such that all entries of
7241:such that all entries of
7153:such that all entries of
6444:is a probability vector,
5106:, one may start with the
4232:. If, by whatever means,
4021:of the transition matrix
2979:{\displaystyle X_{t+h}=j}
1748:conditional probabilities
1359:; for example, the state
766:Examples of Markov chains
756:, starting in the 1950s.
700:studied Markov chains on
678:weak law of large numbers
574:
556:
551:
548:
546:
423:probability distributions
15602:Doléans-Dade exponential
15432:Progressively measurable
15230:CoxâIngersollâRoss (CIR)
13769:(Suppl 4): 21292â21296.
13325:University of St Andrews
12255:Parzen, Emanuel (1962).
11857:. Wiley. p. 46, 47.
11284:Sheldon M. Ross (1996).
11055:Sheldon M. Ross (1996).
10779:
10768:Telescoping Markov chain
10713:Markov chain mixing time
10365:note values, frequency (
10331:
10315:
10205:Markov chain Monte Carlo
9990:of a webpage as used by
9717:matrix population models
9426:topological Markov chain
9092:{\displaystyle \varphi }
9072:{\displaystyle \varphi }
9014:{\displaystyle \varphi }
8990:formed by selecting the
7802:. It can be improved to
4230:PerronâFrobenius theorem
3300:exponential distribution
2863:Infinitesimal definition
419:Markov chain Monte Carlo
293:Law of total probability
288:Conditional independence
177:Exponential distribution
162:Probability distribution
18:Equilibrium distribution
15822:Mathematical statistics
15812:Large deviations theory
15642:Infinitesimal generator
15503:Maximal ergodic theorem
15422:Piecewise-deterministic
15024:Random dynamical system
14889:Markov additive process
14017:(12): 129â131, 449â469.
13840:10.1162/014892699559733
13794:10.1073/pnas.1019454108
13637:Journal of Econometrics
13551:2027/coo.31924001082803
12110:Advances in Mathematics
12103:Spitzer, Frank (1970).
11854:Stochastipoic processes
11851:Joseph L. Doob (1990).
11527:Cramér, Harald (1976).
11331:10.1214/lnms/1196285381
10890:Oxford University Press
10723:Markov decision process
10213:posterior distributions
10184:{\displaystyle \alpha }
9699:models of DNA evolution
9430:subshift of finite type
9412:Subshift of finite type
9406:Subshift of finite type
9356:Markov decision process
9099:are greater than 0 and
8752:conditional probability
8419:For a subset of states
8402:Phase-type distribution
8222:{\displaystyle \Omega }
7989:be the shift operator:
7949:{\displaystyle \Sigma }
7584:{\displaystyle k\geq 1}
7210:, defined according to
6670:greatest common divisor
6634:Two states are said to
4872:) to return the matrix
4655:It is always true that
3852:right stochastic matrix
3648:is the solution of the
2940:{\displaystyle X_{t}=i}
2665:{\displaystyle (X_{n})}
2632:{\displaystyle (Y_{n})}
1840:The possible values of
870:{\displaystyle X_{0}=0}
272:Conditional probability
15921:Random text generation
15657:KarhunenâLoĂšve theorem
15592:CameronâMartin formula
15556:BurkholderâDavisâGundy
14951:Variance gamma process
14239:. John Wiley and Sons.
13828:Computer Music Journal
13687:10.1093/jjfinec/nbh003
12531:Fitzpatrick, Richard.
12343:Mathematics LibreTexts
11546:10.1214/aop/1176996025
11028:John Lamperti (1977).
10646:Markov text generators
10282:
10262:Credit rating agencies
10185:
10165:
10130:
10075:
10048:
10028:
10008:
9983:
9976:
9852:Lempel-Ziv compression
9830:reinforcement learning
9618:
9391:Note, however, by the
9255:
9127:
9093:
9073:
9050:
9015:
8965:
8888:
8701:Kolmogorov's criterion
8695:A chain is said to be
8682:
8610:
8457:
8415:Expected hitting times
8365:
8223:
8203:
8177:
8132:
8098:
8063:
7983:
7950:
7930:
7878:
7858:
7792:
7772:
7725:
7693:
7666:
7623:
7585:
7559:
7536:
7484:
7464:
7412:
7389:
7349:
7329:
7302:
7262:
7235:
7174:
7147:
7104:
7020:
6982:
6944:
6834:
6808:
6807:{\displaystyle k>1}
6775:
6550:
6416:
5547:
5495:
5442:
5343:
5216:
5071:
4973:
4770:
4688:
4647:
4572:
4439:
4323:
4271:
4215:
4109:
4056:
4011:
3965:
3901:
3833:
3726:If the state space is
3705:
3630:
3233:
3232:{\displaystyle q_{ij}}
3195:
3165:
3029:
2980:
2941:
2896:
2872:
2817:transition rate matrix
2774:
2666:
2633:
2596:
2237:
2202:
1979:
1831:
1740:
1495:
1444:
1398:
1353:
1326:
1299:
1253:
1215:
1166:
1135:
1105:
1078:
1051:
1024:
988:
967:. If we know not just
961:
918:
871:
834:
549:Countable state space
532:Types of Markov chains
496:A Markov process is a
488:
484:Russian mathematician
214:Continuous or discrete
167:Bernoulli distribution
35:
15787:Actuarial mathematics
15749:Uniform integrability
15744:Stratonovich integral
15672:LĂ©vyâProkhorov metric
15576:MarcinkiewiczâZygmund
15463:Central limit theorem
15065:Gaussian random field
14894:McKeanâVlasov process
14814:Dyson Brownian motion
14675:GaltonâWatson process
14396:Markovskiye protsessy
13358:U.S. patent 6,285,999
11081:Everitt, B.S. (2002)
10219:Economics and finance
10186:
10166:
10131:
10076:
10074:{\displaystyle k_{i}}
10049:
10029:
10009:
9977:
9926:
9919:Internet applications
9793:'s famous 1948 paper
9619:
9490:statistical mechanics
9352:System is controlled
9339:System is autonomous
9256:
9128:
9094:
9074:
9051:
9016:
8966:
8889:
8750:, and represents the
8707:Embedded Markov chain
8683:
8611:
8458:
8366:
8224:
8204:
8178:
8133:
8099:
8064:
7984:
7951:
7931:
7886:diameter of the graph
7879:
7859:
7793:
7773:
7726:
7694:
7692:{\displaystyle M^{2}}
7667:
7624:
7586:
7560:
7537:
7485:
7465:
7413:
7390:
7350:
7330:
7328:{\displaystyle M^{k}}
7303:
7263:
7261:{\displaystyle M^{k}}
7236:
7175:
7173:{\displaystyle M^{k}}
7148:
7105:
7021:
7019:{\displaystyle M_{i}}
6983:
6924:
6835:
6809:
6776:
6551:
6417:
5548:
5475:
5443:
5344:
5217:
5090:. Then assuming that
5072:
4974:
4771:
4689:
4648:
4573:
4440:
4324:
4272:
4216:
4110:
4057:
4012:
3966:
3902:
3834:
3706:
3631:
3302:with rate parameter â
3234:
3196:
3166:
3030:
2981:
2942:
2897:
2895:{\displaystyle X_{t}}
2870:
2775:
2667:
2634:
2597:
2238:
2236:{\displaystyle X_{0}}
2203:
1980:
1832:
1741:
1496:
1445:
1399:
1354:
1352:{\displaystyle X_{1}}
1327:
1325:{\displaystyle X_{2}}
1300:
1254:
1216:
1167:
1165:{\displaystyle X_{n}}
1136:
1134:{\displaystyle X_{n}}
1106:
1104:{\displaystyle X_{6}}
1079:
1077:{\displaystyle X_{7}}
1052:
1050:{\displaystyle X_{6}}
1025:
989:
987:{\displaystyle X_{6}}
962:
919:
872:
835:
833:{\displaystyle X_{n}}
723:Irénée-Jules Bienaymé
691:central limit theorem
483:
172:Binomial distribution
33:
15862:Time series analysis
15817:Mathematical finance
15702:Reflection principle
15029:Regenerative process
14829:FlemingâViot process
14644:Stochastic processes
14460:Finite Markov Chains
14446:Finite Markov Chains
14333:Majone, Giandomenico
14287:Stochastic Processes
14264:Leo Breiman (1992)
13994:on December 6, 2010.
13476:The Economic Journal
13453:(Technical report).
13311:Robertson, Edmund F.
12477:. pp. 108â127.
12396:Discrete Mathematics
12257:Stochastic Processes
11938:. Berlin: Springer.
11827:Stochastic Processes
11691:10.1112/blms/22.1.31
11511:10.1112/blms/22.1.31
11287:Stochastic processes
11103:Stochastic Processes
11058:Stochastic processes
10977:Stochastic Processes
10753:Quantum Markov chain
10698:GaussâMarkov process
10676:stochastic terrorism
10290:economic development
10175:
10140:
10085:
10058:
10038:
10018:
9998:
9931:
9907: â 1 (for
9775:Hidden Markov models
9735:Compartmental models
9512:
9466:block-coding systems
9462:context-free systems
9145:
9104:
9083:
9063:
9028:
9005:
8908:
8769:
8637:
8482:
8467:, starting in state
8435:
8271:
8229:(up to a null set).
8213:
8187:
8142:
8122:
8073:
7993:
7961:
7940:
7905:
7868:
7806:
7782:
7738:
7703:
7676:
7636:
7595:
7569:
7549:
7494:
7474:
7426:
7402:
7359:
7339:
7312:
7292:
7282:index of primitivity
7276:Index of primitivity
7245:
7225:
7157:
7137:
7057:
7003:
6886:
6818:
6792:
6691:
6491:
5612:
5455:
5420:
5232:
5177:
5043:
5023:th row or column of
4883:
4708:
4662:
4601:
4450:
4336:
4288:
4236:
4169:
4070:
4037:
3978:
3921:
3877:
3757:
3663:
3396:
3389:, ... it holds that
3213:
3175:
3051:
2990:
2951:
2918:
2879:
2684:
2643:
2610:
2258:
2220:
2000:
1869:
1754:
1555:
1454:
1409:
1363:
1336:
1309:
1264:
1225:
1180:
1149:
1118:
1088:
1061:
1034:
998:
971:
935:
881:
877:, then the sequence
848:
817:
808:A non-Markov example
783:, also known as the
725:. Starting in 1928,
718:Henry William Watson
398:process is called a
298:Law of large numbers
267:Marginal probability
192:Poisson distribution
41:Part of a series on
15857:Stochastic analysis
15697:Quadratic variation
15692:Prokhorov's theorem
15627:FeynmanâKac formula
15097:Markov random field
14745:Birthâdeath process
14391:ĐĐ°ŃĐșĐŸĐČŃĐșОД ĐżŃĐŸŃĐ”ŃŃŃ
14203:2019SoEn..184..688M
13881:, Pendragon Press.
13775:2011PNAS..10821292A
13737:Columbia University
13568:Journal of Business
13418:2006SJSC...27.2112L
13309:O'Connor, John J.;
13270:2015NatSR...510203P
13166:2018SoEn..170..174M
13131:1998SoEn...62..101M
13096:2018SoEn..173..487M
13059:2015SoEn..115..229B
13020:2014SoEn..103..160N
12985:1988SoEn...40..269A
12932:2014AIChE..60.1253G
12873:2009PLSCB...5E0532G
12826:2012AcCrA..68..148K
12702:2012arXiv1209.6210D
12074:2011PhRvE..84d1112F
11902:. pp. 60â107.
11628:2005AmJPh..73..395B
11268:10.1511/2013.101.92
10888:(Online ed.).
10758:Semi-Markov process
10743:Markov random field
10457:
10374:
10256:general equilibrium
9874:Agner Krarup Erlang
9826:pattern recognition
9816:techniques such as
9713:Population dynamics
9547:
9346:Hidden Markov model
8576:
8503:
8452:
7672:is symmetric, then
6974:
6833:{\displaystyle k=1}
6580:General state space
6402:
6325:
6254:
6183:
6149:
6084:
6065:
6031:
6012:
5984:
5965:
5895:
5857:
5825:
4597:matrix, and define
500:that satisfies the
427:Bayesian statistics
257:Complementary event
199:Probability measure
187:Pareto distribution
182:Normal distribution
15827:Probability theory
15707:Skorokhod integral
15677:Malliavin calculus
15260:Korn-Kreer-Lenssen
15144:Time series models
15107:PitmanâYor process
14584:2008-05-22 at the
14246:Science in Context
13922:2012-04-14 at the
13746:on March 24, 2016.
13542:10.24033/asens.476
13343:2015-05-13 at the
13336:S. P. Meyn, 2007.
13258:Scientific Reports
12710:10.1214/12-aoas541
11315:. pp. 75â91.
11256:American Scientist
11101:Parzen, E. (1962)
10455:
10372:
10358:probability vector
10340:, particularly in
10322:Snakes and Ladders
10225:D. G. Champernowne
10209:Bayesian inference
10181:
10161:
10126:
10071:
10044:
10024:
10004:
9984:
9972:
9801:information theory
9785:Information theory
9777:have been used in
9770:Speech recognition
9614:
9582:
9566:
9251:
9123:
9089:
9069:
9046:
9011:
8961:
8901:may be written as
8884:
8879:
8827:
8678:
8606:
8604:
8562:
8548:
8489:
8453:
8438:
8361:
8219:
8199:
8173:
8128:
8094:
8059:
7979:
7946:
7926:
7874:
7854:
7800:girth of the graph
7788:
7768:
7721:
7689:
7662:
7619:
7581:
7555:
7532:
7480:
7460:
7408:
7385:
7345:
7325:
7298:
7258:
7231:
7170:
7143:
7100:
7016:
6997:positive recurrent
6978:
6951:
6855:if, starting from
6830:
6804:
6771:
6546:
6412:
6410:
6384:
6307:
6236:
6165:
6135:
6066:
6051:
6013:
5998:
5966:
5951:
5877:
5839:
5807:
5570:. In other words,
5543:
5438:
5393:corresponding to λ
5339:
5212:
5169:eigendecomposition
5108:Jordan normal form
5104:defective matrices
5067:
4969:
4766:
4684:
4643:
4627:
4568:
4562:
4521:
4485:
4435:
4377:
4319:
4306:
4267:
4254:
4211:
4187:
4105:
4082:
4052:
4051:
4007:
3990:
3961:
3945:
3897:
3842:Since each row of
3829:
3722:Finite state space
3701:
3626:
3229:
3191:
3161:
3025:
2976:
2937:
2892:
2873:
2770:
2662:
2629:
2592:
2590:
2233:
2198:
1975:
1827:
1736:
1491:
1440:
1394:
1349:
1322:
1295:
1249:
1211:
1162:
1131:
1101:
1074:
1047:
1020:
984:
957:
928:a Markov process.
914:
867:
830:
617:countably infinite
498:stochastic process
489:
447:information theory
415:statistical models
388:countably infinite
372:stochastic process
308:Boole's inequality
244:Stochastic process
133:Mutual exclusivity
50:Probability theory
36:
15893:
15892:
15847:Signal processing
15566:Doob's upcrossing
15561:Doob's martingale
15525:EngelbertâSchmidt
15468:Donsker's theorem
15402:Feller-continuous
15270:RendlemanâBartter
15060:Dirichlet process
14977:Branching process
14946:Telegraph process
14839:Geometric process
14819:Empirical process
14809:Diffusion process
14665:Branching process
14660:Bernoulli process
14546:978-0-7923-9650-5
14497:Kishor S. Trivedi
14492:978-0-387-29765-1
14415:978-0-521-88441-9
14386:. Title-No. 5105.
14383:978-3-662-23320-7
14357:. Title-No. 5104.
14354:978-3-662-00033-5
14279:. (See Chapter 7)
14043:978-0-8195-2239-9
13905:on July 13, 2012.
13864:978-0-262-18158-7
13436:10.1137/040607551
13383:978-1-5225-0106-0
13278:10.1038/srep10203
12940:10.1002/aic.14409
12820:(Pt 1): 148â155.
12790:10.1021/ci9000458
12755:10.1021/ci9000458
12628:978-3-642-01849-7
12434:978-0-387-95313-7
12376:978-0-470-77605-6
12324:978-0-470-77605-6
12062:Physical Review E
11953:978-3-540-89331-8
11881:978-1-4612-3166-0
11837:978-0-486-79688-8
11810:978-0-444-86937-1
11750:978-0-08-057041-9
11720:978-1-118-59320-2
11662:978-1-4612-3038-0
11636:10.1119/1.1848117
11600:978-3-540-26312-8
11573:978-1-4471-7262-8
11477:978-0-387-95283-3
11340:978-0-940600-61-4
11297:978-0-471-12062-9
11235:978-1-4757-3124-8
11203:978-0-8218-0749-1
11122:Dodge, Y. (2003)
11068:978-0-471-12062-9
11041:978-3-540-90275-1
11014:978-0-08-057041-9
10987:978-0-486-79688-8
10960:978-0-387-00211-8
10896:(Subscription or
10863:978-1-58488-587-0
10836:978-1-118-21052-9
10809:978-0-521-73182-9
10662:, Jeff Harrison,
10660:dissociated press
10600:
10599:
10456:2nd-order matrix
10453:
10452:
10373:1st-order matrix
10310:democratic regime
10249:Laurent E. Calvet
10241:James D. Hamilton
10195:individual user.
10159:
10124:
10103:
10047:{\displaystyle i}
10027:{\displaystyle N}
10007:{\displaystyle i}
9970:
9949:
9837:Viterbi algorithm
9818:arithmetic coding
9612:
9605:
9604:
9603:
9600:
9595:
9589:
9581:
9580:
9577:
9572:
9568:
9529:
9525:
9519:
9438:dynamical systems
9386:Bernoulli process
9366:
9365:
9246:
8872:
8849:
8842:
8812:
8650:
8588:
8533:
8515:
8131:{\displaystyle S}
7877:{\displaystyle d}
7791:{\displaystyle s}
7558:{\displaystyle M}
7527:
7483:{\displaystyle M}
7411:{\displaystyle n}
7348:{\displaystyle M}
7301:{\displaystyle k}
7234:{\displaystyle k}
7208:ergodic processes
7146:{\displaystyle k}
6840:and the state is
6372:
6295:
6224:
6114:
5380:matrix, that is,
5085:stochastic matrix
4987:stochastic matrix
4827:stochastic matrix
4612:
4558:
4546:
4481:
4469:
4291:
4239:
4172:
4073:
3981:
3974:is a normalized (
3959:
3936:
3207:little-o notation
2914:. Then, knowing
2577:
1505:Formal definition
1172:to represent the
1141:to represent the
787:process, and the
733:Andrey Kolmogorov
710:Tatyana Ehrenfest
693:for such chains.
687:Alexander Pushkin
633:transition matrix
593:
592:
459:speech processing
455:signal processing
360:
359:
262:Joint probability
209:Bernoulli process
108:Probability space
16:(Redirected from
15928:
15906:Markov processes
15867:Machine learning
15754:Usual hypotheses
15637:Girsanov theorem
15622:Dynkin's formula
15387:Continuous paths
15295:Actuarial models
15235:GarmanâKohlhagen
15205:BlackâKarasinski
15200:BlackâDermanâToy
15187:Financial models
15053:Fields and other
14982:Gaussian process
14931:Sigma-martingale
14735:Additive process
14637:
14630:
14623:
14614:
14613:
14593:
14575:
14443:
14441:
14429:
14393:
14392:
14387:
14367:
14364:Markov Processes
14358:
14330:
14327:Markov Processes
14261:
14215:
14214:
14186:
14177:
14176:
14174:
14172:
14163:
14154:
14148:
14147:
14145:
14143:
14137:10.1108/eb022949
14116:
14110:
14109:
14107:
14083:
14074:
14073:
14066:Energy Economics
14057:
14048:
14047:
14035:
14025:
14019:
14018:
14007:O'Rourke, Joseph
14002:
13996:
13995:
13990:. Archived from
13984:
13978:
13977:
13975:
13974:
13966:Pankin, Mark D.
13963:
13957:
13956:
13954:
13953:
13944:. Archived from
13940:Pankin, Mark D.
13937:
13931:
13913:
13907:
13906:
13901:. Archived from
13895:
13889:
13875:
13869:
13868:
13850:
13844:
13843:
13823:
13817:
13816:
13806:
13796:
13786:
13754:
13748:
13747:
13745:
13739:. Archived from
13734:
13726:
13720:
13719:
13717:
13711:. Archived from
13706:
13697:
13691:
13690:
13680:
13660:
13654:
13653:
13627:
13621:
13620:
13602:
13582:
13576:
13575:
13562:
13556:
13555:
13553:
13525:
13519:
13518:
13506:
13500:
13499:
13471:
13465:
13464:
13462:
13446:
13440:
13439:
13429:
13412:(6): 2112â2113.
13403:
13394:
13388:
13387:
13367:
13361:
13360:
13354:
13348:
13334:
13328:
13327:
13306:
13300:
13299:
13289:
13264:(10203): 10203.
13249:
13243:
13242:
13223:
13217:
13216:
13199:(3): 1429â1448.
13184:
13178:
13177:
13149:
13143:
13142:
13114:
13108:
13107:
13079:
13073:
13072:
13070:
13038:
13032:
13031:
13003:
12997:
12996:
12968:
12962:
12961:
12951:
12926:(4): 1253â1268.
12911:
12905:
12904:
12894:
12884:
12867:(10): e1000532.
12861:PLOS Comput Biol
12852:
12846:
12845:
12808:
12802:
12801:
12784:(7): 1630â1642.
12773:
12767:
12766:
12749:(7): 1630â1642.
12738:
12732:
12731:
12721:
12695:
12671:
12665:
12664:
12639:
12633:
12632:
12604:
12598:
12597:
12587:
12555:
12546:
12545:
12543:
12542:
12537:
12528:
12522:
12508:
12497:
12496:
12467:
12461:
12460:
12454:
12445:
12439:
12438:
12420:
12414:
12413:
12411:
12387:
12381:
12380:
12360:
12354:
12353:
12351:
12350:
12335:
12329:
12328:
12308:
12302:
12301:
12299:
12298:
12288:"Ergodic Theory"
12280:
12271:
12270:
12252:
12246:
12245:
12243:
12241:
12236:
12224:
12218:
12217:
12215:
12214:
12196:
12190:
12189:
12171:
12165:
12164:
12162:
12161:
12139:Dobrushin, R. L.
12135:
12129:
12128:
12126:
12100:
12094:
12093:
12057:
12051:
12050:
12048:
12047:
12012:
12006:
12005:
12003:
11987:
11981:
11980:
11978:
11977:
11972:
11964:
11958:
11957:
11931:
11922:
11921:
11892:
11886:
11885:
11865:
11859:
11858:
11848:
11842:
11841:
11821:
11815:
11814:
11794:
11788:
11787:
11761:
11755:
11754:
11734:
11725:
11724:
11704:
11695:
11694:
11673:
11667:
11666:
11646:
11640:
11639:
11611:
11605:
11604:
11584:
11578:
11577:
11557:
11551:
11550:
11548:
11524:
11515:
11514:
11493:
11482:
11481:
11451:
11445:
11444:
11416:
11410:
11409:
11378:
11372:
11371:
11351:
11345:
11344:
11324:
11308:
11302:
11301:
11281:
11272:
11271:
11251:
11240:
11239:
11219:
11208:
11207:
11183:
11168:
11161:
11155:
11141:
11135:
11120:
11114:
11099:
11093:
11079:
11073:
11072:
11052:
11046:
11045:
11025:
11019:
11018:
10998:
10992:
10991:
10971:
10965:
10964:
10944:
10935:
10934:
10908:
10902:
10901:
10893:
10881:
10874:
10868:
10867:
10847:
10841:
10840:
10820:
10814:
10813:
10793:
10680:solar irradiance
10658:" software (see
10656:parody generator
10458:
10454:
10440:
10439:
10421:
10420:
10397:
10396:
10389:
10388:
10375:
10371:
10287:
10229:Herbert A. Simon
10190:
10188:
10187:
10182:
10170:
10168:
10167:
10162:
10160:
10155:
10144:
10135:
10133:
10132:
10127:
10125:
10120:
10109:
10104:
10102:
10101:
10089:
10080:
10078:
10077:
10072:
10070:
10069:
10053:
10051:
10050:
10045:
10033:
10031:
10030:
10025:
10013:
10011:
10010:
10005:
9981:
9979:
9978:
9973:
9971:
9966:
9955:
9950:
9948:
9947:
9935:
9822:state estimation
9814:entropy encoding
9810:data compression
9759:Solar irradiance
9623:
9621:
9620:
9615:
9613:
9610:
9606:
9601:
9598:
9596:
9593:
9587:
9585:
9583:
9578:
9575:
9573:
9570:
9569:
9567:
9565:
9564:
9557:
9549:
9548:
9546:
9539:
9531:
9527:
9520:
9517:
9446:closed manifolds
9418:adjacency matrix
9381:Bernoulli scheme
9375:Bernoulli scheme
9369:Bernoulli scheme
9326:
9325:
9275:
9260:
9258:
9257:
9252:
9247:
9245:
9244:
9239:
9235:
9234:
9233:
9194:
9193:
9192:
9155:
9138:may be found as
9137:
9134:= 1. From this,
9132:
9130:
9129:
9124:
9122:
9121:
9098:
9096:
9095:
9090:
9078:
9076:
9075:
9070:
9055:
9053:
9052:
9047:
9020:
9018:
9017:
9012:
8994:from the matrix
8970:
8968:
8967:
8962:
8957:
8956:
8948:
8944:
8893:
8891:
8890:
8885:
8883:
8882:
8873:
8870:
8850:
8847:
8843:
8841:
8840:
8839:
8826:
8810:
8809:
8797:
8784:
8783:
8741:, is denoted by
8718:
8687:
8685:
8684:
8679:
8677:
8676:
8658:
8657:
8652:
8651:
8643:
8615:
8613:
8612:
8607:
8605:
8589:
8586:
8575:
8570:
8561:
8560:
8547:
8516:
8513:
8502:
8497:
8462:
8460:
8459:
8454:
8451:
8446:
8370:
8368:
8367:
8362:
8357:
8356:
8295:
8294:
8228:
8226:
8225:
8220:
8208:
8206:
8205:
8200:
8182:
8180:
8179:
8174:
8157:
8156:
8137:
8135:
8134:
8129:
8103:
8101:
8100:
8095:
8093:
8092:
8091:
8068:
8066:
8065:
8060:
8049:
8048:
8024:
8023:
8011:
8010:
7988:
7986:
7985:
7980:
7955:
7953:
7952:
7947:
7935:
7933:
7932:
7927:
7925:
7924:
7923:
7883:
7881:
7880:
7875:
7863:
7861:
7860:
7855:
7797:
7795:
7794:
7789:
7777:
7775:
7774:
7769:
7730:
7728:
7727:
7722:
7698:
7696:
7695:
7690:
7688:
7687:
7671:
7669:
7668:
7663:
7652:
7628:
7626:
7625:
7620:
7590:
7588:
7587:
7582:
7564:
7562:
7561:
7556:
7541:
7539:
7538:
7533:
7528:
7525:
7489:
7487:
7486:
7481:
7469:
7467:
7466:
7461:
7453:
7452:
7422:The exponent is
7417:
7415:
7414:
7409:
7394:
7392:
7391:
7386:
7375:
7354:
7352:
7351:
7346:
7334:
7332:
7331:
7326:
7324:
7323:
7307:
7305:
7304:
7299:
7267:
7265:
7264:
7259:
7257:
7256:
7240:
7238:
7237:
7232:
7195: = 1.
7179:
7177:
7176:
7171:
7169:
7168:
7152:
7150:
7149:
7144:
7109:
7107:
7106:
7101:
7096:
7095:
7080:
7069:
7068:
7025:
7023:
7022:
7017:
7015:
7014:
6987:
6985:
6984:
6979:
6973:
6962:
6943:
6938:
6917:
6916:
6898:
6897:
6839:
6837:
6836:
6831:
6813:
6811:
6810:
6805:
6780:
6778:
6777:
6772:
6752:
6751:
6733:
6732:
6683:
6677:
6667:
6661:
6655:
6574:
6555:
6553:
6552:
6547:
6542:
6537:
6536:
6527:
6513:
6508:
6507:
6498:
6467:
6448:
6442:
6429:
6421:
6419:
6418:
6413:
6411:
6407:
6403:
6401:
6400:
6394:
6389:
6383:
6382:
6377:
6373:
6371:
6370:
6361:
6360:
6351:
6344:
6343:
6324:
6323:
6317:
6312:
6306:
6305:
6300:
6296:
6294:
6293:
6284:
6283:
6274:
6267:
6266:
6253:
6252:
6246:
6241:
6235:
6234:
6229:
6225:
6223:
6222:
6213:
6212:
6203:
6196:
6195:
6182:
6181:
6175:
6170:
6164:
6163:
6148:
6143:
6128:
6115:
6112:
6110:
6109:
6097:
6096:
6086:
6083:
6082:
6076:
6071:
6064:
6059:
6050:
6049:
6030:
6029:
6023:
6018:
6011:
6006:
5997:
5996:
5983:
5982:
5976:
5971:
5964:
5959:
5950:
5949:
5934:
5930:
5929:
5921:
5915:
5914:
5909:
5900:
5896:
5894:
5893:
5887:
5882:
5876:
5875:
5856:
5855:
5849:
5844:
5838:
5837:
5824:
5823:
5817:
5812:
5806:
5805:
5785:
5781:
5780:
5772:
5766:
5765:
5760:
5742:
5738:
5734:
5733:
5725:
5706:
5702:
5701:
5693:
5677:
5673:
5672:
5664:
5648:
5636:
5635:
5624:
5605:â â. That means
5574:
5568:
5552:
5550:
5549:
5544:
5539:
5531:
5530:
5517:
5516:
5511:
5505:
5504:
5494:
5489:
5471:
5470:
5469:
5463:
5447:
5445:
5444:
5439:
5434:
5433:
5428:
5361:
5348:
5346:
5345:
5340:
5335:
5330:
5329:
5320:
5306:
5301:
5300:
5291:
5283:
5278:
5277:
5268:
5260:
5255:
5254:
5245:
5221:
5219:
5218:
5213:
5208:
5207:
5199:
5184:
5081:
5076:
5074:
5073:
5068:
5063:
5058:
5050:
4978:
4976:
4975:
4970:
4965:
4964:
4949:
4948:
4943:
4934:
4917:
4916:
4905:
4890:
4841:is 1, there are
4775:
4773:
4772:
4767:
4762:
4761:
4750:
4738:
4737:
4732:
4723:
4715:
4693:
4691:
4690:
4685:
4680:
4672:
4652:
4650:
4649:
4644:
4639:
4638:
4633:
4626:
4608:
4577:
4575:
4574:
4569:
4567:
4566:
4559:
4551:
4547:
4539:
4526:
4525:
4490:
4489:
4482:
4474:
4470:
4462:
4444:
4442:
4441:
4436:
4434:
4426:
4425:
4411:
4398:
4397:
4389:
4382:
4381:
4343:
4328:
4326:
4325:
4320:
4318:
4317:
4312:
4305:
4276:
4274:
4273:
4268:
4266:
4265:
4260:
4253:
4220:
4218:
4217:
4212:
4207:
4199:
4198:
4193:
4186:
4161:
4153:
4115:we see that the
4114:
4112:
4111:
4106:
4098:
4097:
4081:
4061:
4059:
4058:
4053:
4050:
4049:
4016:
4014:
4013:
4008:
4000:
3999:
3989:
3970:
3968:
3967:
3962:
3960:
3958:
3957:
3956:
3955:
3944:
3931:
3906:
3904:
3903:
3898:
3887:
3865:
3838:
3836:
3835:
3830:
3816:
3815:
3797:
3796:
3772:
3771:
3710:
3708:
3707:
3702:
3673:
3650:forward equation
3635:
3633:
3632:
3627:
3622:
3621:
3609:
3608:
3590:
3589:
3588:
3587:
3572:
3571:
3551:
3550:
3538:
3537:
3536:
3535:
3512:
3511:
3499:
3498:
3497:
3496:
3479:
3478:
3466:
3465:
3464:
3463:
3446:
3445:
3427:
3426:
3425:
3424:
3264:to describe the
3238:
3236:
3235:
3230:
3228:
3227:
3200:
3198:
3197:
3192:
3190:
3189:
3170:
3168:
3167:
3162:
3139:
3138:
3123:
3122:
3034:
3032:
3031:
3026:
3024:
3020:
3007:
3006:
2985:
2983:
2982:
2977:
2969:
2968:
2946:
2944:
2943:
2938:
2930:
2929:
2901:
2899:
2898:
2893:
2891:
2890:
2779:
2777:
2776:
2771:
2769:
2765:
2764:
2763:
2733:
2732:
2714:
2713:
2696:
2695:
2671:
2669:
2668:
2663:
2658:
2657:
2638:
2636:
2635:
2630:
2625:
2624:
2601:
2599:
2598:
2593:
2591:
2578:
2575:
2570:
2569:
2551:
2550:
2526:
2525:
2507:
2506:
2488:
2487:
2469:
2468:
2450:
2449:
2437:
2436:
2407:
2406:
2394:
2393:
2375:
2374:
2356:
2355:
2337:
2336:
2318:
2317:
2299:
2298:
2286:
2285:
2266:
2242:
2240:
2239:
2234:
2232:
2231:
2207:
2205:
2204:
2199:
2194:
2193:
2181:
2180:
2156:
2155:
2143:
2142:
2124:
2123:
2111:
2110:
2089:
2088:
2076:
2075:
2057:
2056:
2044:
2043:
2031:
2030:
2018:
2017:
1984:
1982:
1981:
1976:
1965:
1964:
1940:
1939:
1912:
1911:
1893:
1892:
1836:
1834:
1833:
1828:
1817:
1816:
1804:
1803:
1785:
1784:
1772:
1771:
1745:
1743:
1742:
1737:
1729:
1728:
1716:
1715:
1697:
1696:
1669:
1668:
1656:
1655:
1637:
1636:
1624:
1623:
1611:
1610:
1598:
1597:
1579:
1578:
1522:random variables
1500:
1498:
1497:
1492:
1472:
1471:
1449:
1447:
1446:
1441:
1421:
1420:
1403:
1401:
1400:
1395:
1375:
1374:
1358:
1356:
1355:
1350:
1348:
1347:
1331:
1329:
1328:
1323:
1321:
1320:
1304:
1302:
1301:
1296:
1276:
1275:
1258:
1256:
1255:
1250:
1220:
1218:
1217:
1212:
1192:
1191:
1171:
1169:
1168:
1163:
1161:
1160:
1140:
1138:
1137:
1132:
1130:
1129:
1110:
1108:
1107:
1102:
1100:
1099:
1083:
1081:
1080:
1075:
1073:
1072:
1056:
1054:
1053:
1048:
1046:
1045:
1029:
1027:
1026:
1021:
1010:
1009:
993:
991:
990:
985:
983:
982:
966:
964:
963:
958:
947:
946:
923:
921:
920:
915:
910:
896:
895:
876:
874:
873:
868:
860:
859:
843:
839:
837:
836:
831:
829:
828:
575:Continuous-time
544:
543:
352:
345:
338:
128:Elementary event
60:
38:
37:
21:
15936:
15935:
15931:
15930:
15929:
15927:
15926:
15925:
15896:
15895:
15894:
15889:
15871:
15832:Queueing theory
15775:
15717:Skorokhod space
15580:
15571:KunitaâWatanabe
15542:
15508:Sanov's theorem
15478:Ergodic theorem
15451:
15447:Time-reversible
15365:
15328:Queueing models
15322:
15318:SparreâAnderson
15308:CramĂ©râLundberg
15289:
15275:SABR volatility
15181:
15138:
15090:Boolean network
15048:
15034:Renewal process
14965:
14914:Non-homogeneous
14904:Poisson process
14794:Contact process
14757:Brownian motion
14727:Continuous time
14721:
14715:Maximal entropy
14646:
14641:
14611:
14591:
14586:Wayback Machine
14560:
14556:
14551:
14456:J. Laurie Snell
14448:pp. 384ff.
14384:
14355:
14337:Springer-Verlag
14224:
14219:
14218:
14187:
14180:
14170:
14168:
14161:
14155:
14151:
14141:
14139:
14117:
14113:
14084:
14077:
14058:
14051:
14044:
14026:
14022:
14003:
13999:
13986:
13985:
13981:
13972:
13970:
13964:
13960:
13951:
13949:
13938:
13934:
13924:Wayback Machine
13914:
13910:
13897:
13896:
13892:
13876:
13872:
13865:
13851:
13847:
13824:
13820:
13784:10.1.1.225.6090
13755:
13751:
13743:
13732:
13728:
13727:
13723:
13715:
13704:
13698:
13694:
13678:10.1.1.536.8334
13661:
13657:
13628:
13624:
13609:10.2307/1912559
13600:10.1.1.397.3582
13583:
13579:
13563:
13559:
13526:
13522:
13507:
13503:
13488:10.2307/2227127
13482:(250): 318â51.
13472:
13468:
13447:
13443:
13401:
13395:
13391:
13384:
13368:
13364:
13356:
13355:
13351:
13345:Wayback Machine
13335:
13331:
13307:
13303:
13250:
13246:
13224:
13220:
13185:
13181:
13150:
13146:
13115:
13111:
13080:
13076:
13039:
13035:
13004:
13000:
12969:
12965:
12912:
12908:
12853:
12849:
12809:
12805:
12774:
12770:
12739:
12735:
12672:
12668:
12662:
12640:
12636:
12629:
12605:
12601:
12556:
12549:
12540:
12538:
12535:
12529:
12525:
12509:
12500:
12493:
12468:
12464:
12452:
12446:
12442:
12435:
12421:
12417:
12388:
12384:
12377:
12361:
12357:
12348:
12346:
12337:
12336:
12332:
12325:
12309:
12305:
12296:
12294:
12281:
12274:
12267:
12253:
12249:
12239:
12237:
12234:
12225:
12221:
12212:
12210:
12197:
12193:
12186:
12172:
12168:
12159:
12157:
12155:
12136:
12132:
12101:
12097:
12058:
12054:
12045:
12043:
12033:10.1137/1037083
12013:
12009:
11988:
11984:
11975:
11973:
11970:
11966:
11965:
11961:
11954:
11932:
11925:
11918:
11893:
11889:
11882:
11866:
11862:
11849:
11845:
11838:
11822:
11818:
11811:
11795:
11791:
11784:
11762:
11758:
11751:
11735:
11728:
11721:
11705:
11698:
11674:
11670:
11663:
11647:
11643:
11612:
11608:
11601:
11585:
11581:
11574:
11558:
11554:
11525:
11518:
11494:
11485:
11478:
11452:
11448:
11433:10.2307/1403518
11417:
11413:
11398:10.2307/1403785
11379:
11375:
11352:
11348:
11341:
11309:
11305:
11298:
11282:
11275:
11252:
11243:
11236:
11220:
11211:
11204:
11184:
11171:
11162:
11158:
11142:
11138:
11121:
11117:
11100:
11096:
11080:
11076:
11069:
11053:
11049:
11042:
11026:
11022:
11015:
10999:
10995:
10988:
10972:
10968:
10961:
10945:
10938:
10923:
10909:
10905:
10895:
10875:
10871:
10864:
10848:
10844:
10837:
10821:
10817:
10810:
10794:
10787:
10782:
10777:
10748:Master equation
10738:Markov operator
10733:Markov odometer
10688:
10672:
10648:
10627:
10437:
10436:
10418:
10417:
10394:
10393:
10386:
10385:
10334:
10326:Hi Ho! Cherry-O
10318:
10292:to the rise of
10270:
10268:Social sciences
10233:Louis Bachelier
10221:
10201:
10176:
10173:
10172:
10145:
10143:
10141:
10138:
10137:
10110:
10108:
10097:
10093:
10088:
10086:
10083:
10082:
10065:
10061:
10059:
10056:
10055:
10039:
10036:
10035:
10019:
10016:
10015:
9999:
9996:
9995:
9956:
9954:
9943:
9939:
9934:
9932:
9929:
9928:
9921:
9897:Poisson process
9895:according to a
9870:queueing theory
9866:
9864:Queueing theory
9860:
9858:Queueing theory
9787:
9772:
9756:
9744:
9729:Systems biology
9684:
9632:
9631:
9630:
9624:
9586:
9584:
9560:
9553:
9552:
9550:
9542:
9535:
9533:
9532:
9530:
9526:
9524:
9516:
9515:
9513:
9510:
9509:
9505:
9482:
9474:
9442:diffeomorphisms
9414:
9408:
9377:
9371:
9320:
9314:
9309:
9273:
9240:
9226:
9222:
9200:
9196:
9195:
9185:
9181:
9156:
9154:
9146:
9143:
9142:
9135:
9117:
9113:
9105:
9102:
9101:
9084:
9081:
9080:
9064:
9061:
9060:
9029:
9026:
9025:
9006:
9003:
9002:
8988:diagonal matrix
8980:identity matrix
8949:
8928:
8924:
8923:
8909:
8906:
8905:
8878:
8877:
8869:
8867:
8861:
8860:
8846:
8844:
8832:
8828:
8816:
8811:
8802:
8798:
8796:
8789:
8788:
8776:
8772:
8770:
8767:
8766:
8749:
8716:
8709:
8666:
8662:
8653:
8642:
8641:
8640:
8638:
8635:
8634:
8632:
8622:
8603:
8602:
8587: for
8585:
8583:
8571:
8566:
8553:
8549:
8537:
8527:
8526:
8514: for
8512:
8510:
8498:
8493:
8485:
8483:
8480:
8479:
8463:represents the
8447:
8442:
8436:
8433:
8432:
8417:
8404:
8398:
8352:
8351:
8290:
8289:
8272:
8269:
8268:
8246:
8214:
8211:
8210:
8188:
8185:
8184:
8149:
8145:
8143:
8140:
8139:
8123:
8120:
8119:
8087:
8086:
8082:
8074:
8071:
8070:
8044:
8040:
8019:
8015:
8006:
8002:
7994:
7991:
7990:
7962:
7959:
7958:
7941:
7938:
7937:
7919:
7918:
7914:
7906:
7903:
7902:
7895:
7869:
7866:
7865:
7807:
7804:
7803:
7783:
7780:
7779:
7739:
7736:
7735:
7704:
7701:
7700:
7683:
7679:
7677:
7674:
7673:
7639:
7637:
7634:
7633:
7596:
7593:
7592:
7570:
7567:
7566:
7550:
7547:
7546:
7526: and
7524:
7495:
7492:
7491:
7475:
7472:
7471:
7448:
7444:
7427:
7424:
7423:
7403:
7400:
7399:
7362:
7360:
7357:
7356:
7340:
7337:
7336:
7319:
7315:
7313:
7310:
7309:
7293:
7290:
7289:
7278:
7252:
7248:
7246:
7243:
7242:
7226:
7223:
7222:
7204:
7164:
7160:
7158:
7155:
7154:
7138:
7135:
7134:
7116:
7091:
7087:
7076:
7064:
7060:
7058:
7055:
7054:
7048:
7010:
7006:
7004:
7001:
7000:
6963:
6955:
6939:
6928:
6912:
6908:
6893:
6889:
6887:
6884:
6883:
6879:is defined as:
6863:. It is called
6819:
6816:
6815:
6793:
6790:
6789:
6747:
6743:
6728:
6724:
6692:
6689:
6688:
6679:
6673:
6663:
6657:
6651:
6632:
6608:
6593:
6588:
6582:
6572:
6569:
6562:
6538:
6532:
6528:
6523:
6509:
6503:
6499:
6494:
6492:
6489:
6488:
6486:
6479:
6465:
6462:
6456:
6446:
6440:
6437:
6431:is parallel to
6427:
6409:
6408:
6396:
6395:
6390:
6385:
6378:
6366:
6362:
6356:
6352:
6350:
6346:
6345:
6339:
6335:
6319:
6318:
6313:
6308:
6301:
6289:
6285:
6279:
6275:
6273:
6269:
6268:
6262:
6258:
6248:
6247:
6242:
6237:
6230:
6218:
6214:
6208:
6204:
6202:
6198:
6197:
6191:
6187:
6177:
6176:
6171:
6166:
6159:
6155:
6154:
6150:
6144:
6139:
6126:
6125:
6113: for
6111:
6105:
6101:
6092:
6088:
6085:
6078:
6077:
6072:
6067:
6060:
6055:
6045:
6041:
6025:
6024:
6019:
6014:
6007:
6002:
5992:
5988:
5978:
5977:
5972:
5967:
5960:
5955:
5945:
5941:
5932:
5931:
5922:
5917:
5916:
5910:
5902:
5901:
5889:
5888:
5883:
5878:
5871:
5867:
5851:
5850:
5845:
5840:
5833:
5829:
5819:
5818:
5813:
5808:
5801:
5797:
5796:
5792:
5783:
5782:
5773:
5768:
5767:
5761:
5750:
5749:
5740:
5739:
5726:
5715:
5714:
5710:
5694:
5683:
5682:
5678:
5665:
5654:
5653:
5649:
5644:
5637:
5625:
5620:
5619:
5615:
5613:
5610:
5609:
5588:
5582:
5572:
5566:
5556:If we multiply
5535:
5526:
5522:
5512:
5507:
5506:
5500:
5496:
5490:
5479:
5465:
5464:
5459:
5458:
5456:
5453:
5452:
5429:
5424:
5423:
5421:
5418:
5417:
5415:
5398:
5388:
5371:
5359:
5331:
5325:
5321:
5316:
5302:
5296:
5292:
5287:
5279:
5273:
5269:
5264:
5256:
5250:
5246:
5241:
5233:
5230:
5229:
5200:
5189:
5188:
5180:
5178:
5175:
5174:
5166:
5157:
5150:
5143:
5079:
5059:
5054:
5046:
5044:
5041:
5040:
5037:
5010:
4998:has an element
4957:
4953:
4944:
4939:
4938:
4930:
4906:
4901:
4900:
4886:
4884:
4881:
4880:
4808:
4790:identity matrix
4787:
4751:
4746:
4745:
4733:
4728:
4727:
4719:
4711:
4709:
4706:
4705:
4676:
4665:
4663:
4660:
4659:
4634:
4629:
4628:
4616:
4604:
4602:
4599:
4598:
4561:
4560:
4550:
4548:
4538:
4531:
4530:
4520:
4519:
4514:
4508:
4507:
4502:
4492:
4491:
4484:
4483:
4473:
4471:
4461:
4454:
4453:
4451:
4448:
4447:
4430:
4412:
4407:
4406:
4390:
4385:
4384:
4376:
4375:
4370:
4364:
4363:
4358:
4348:
4347:
4339:
4337:
4334:
4333:
4313:
4308:
4307:
4295:
4289:
4286:
4285:
4261:
4256:
4255:
4243:
4237:
4234:
4233:
4203:
4194:
4189:
4188:
4176:
4170:
4167:
4166:
4159:
4151:
4129:
4093:
4089:
4077:
4071:
4068:
4067:
4045:
4041:
4038:
4035:
4034:
3995:
3991:
3985:
3979:
3976:
3975:
3951:
3947:
3946:
3940:
3935:
3930:
3922:
3919:
3918:
3883:
3878:
3875:
3874:
3863:
3860:
3811:
3807:
3786:
3782:
3764:
3760:
3758:
3755:
3754:
3724:
3716:identity matrix
3666:
3664:
3661:
3660:
3647:
3617:
3613:
3598:
3594:
3577:
3573:
3567:
3563:
3562:
3558:
3546:
3542:
3531:
3527:
3526:
3522:
3507:
3503:
3492:
3488:
3487:
3483:
3474:
3470:
3459:
3455:
3454:
3450:
3435:
3431:
3414:
3410:
3409:
3405:
3397:
3394:
3393:
3388:
3381:
3374:
3367:
3360:
3353:
3346:
3330:
3323:
3322:
3314:
3297:
3288:
3281:
3274:
3263:
3253:
3220:
3216:
3214:
3211:
3210:
3203:Kronecker delta
3182:
3178:
3176:
3173:
3172:
3131:
3127:
3115:
3111:
3052:
3049:
3048:
3002:
2998:
2997:
2993:
2991:
2988:
2987:
2958:
2954:
2952:
2949:
2948:
2925:
2921:
2919:
2916:
2915:
2886:
2882:
2880:
2877:
2876:
2865:
2855:
2847:. The elements
2838:
2830:, the elements
2810:
2803:
2793:
2787:
2747:
2743:
2722:
2718:
2709:
2705:
2704:
2700:
2691:
2687:
2685:
2682:
2681:
2653:
2649:
2644:
2641:
2640:
2620:
2616:
2611:
2608:
2607:
2589:
2588:
2576: for
2574:
2559:
2555:
2540:
2536:
2515:
2511:
2496:
2492:
2477:
2473:
2458:
2454:
2445:
2441:
2432:
2428:
2418:
2412:
2411:
2402:
2398:
2389:
2385:
2364:
2360:
2345:
2341:
2326:
2322:
2307:
2303:
2294:
2290:
2281:
2277:
2267:
2265:
2261:
2259:
2256:
2255:
2227:
2223:
2221:
2218:
2217:
2189:
2185:
2170:
2166:
2151:
2147:
2132:
2128:
2119:
2115:
2106:
2102:
2084:
2080:
2071:
2067:
2052:
2048:
2039:
2035:
2026:
2022:
2013:
2009:
2001:
1998:
1997:
1954:
1950:
1935:
1931:
1907:
1903:
1882:
1878:
1870:
1867:
1866:
1862:
1848:
1812:
1808:
1799:
1795:
1780:
1776:
1767:
1763:
1755:
1752:
1751:
1724:
1720:
1711:
1707:
1686:
1682:
1664:
1660:
1651:
1647:
1632:
1628:
1619:
1615:
1606:
1602:
1593:
1589:
1568:
1564:
1556:
1553:
1552:
1546:Markov property
1544:, ... with the
1543:
1536:
1529:
1518:
1512:
1507:
1461:
1457:
1455:
1452:
1451:
1416:
1412:
1410:
1407:
1406:
1370:
1366:
1364:
1361:
1360:
1343:
1339:
1337:
1334:
1333:
1332:now depends on
1316:
1312:
1310:
1307:
1306:
1271:
1267:
1265:
1262:
1261:
1226:
1223:
1222:
1187:
1183:
1181:
1178:
1177:
1156:
1152:
1150:
1147:
1146:
1125:
1121:
1119:
1116:
1115:
1095:
1091:
1089:
1086:
1085:
1068:
1064:
1062:
1059:
1058:
1041:
1037:
1035:
1032:
1031:
1005:
1001:
999:
996:
995:
978:
974:
972:
969:
968:
942:
938:
936:
933:
932:
906:
891:
887:
882:
879:
878:
855:
851:
849:
846:
845:
841:
824:
820:
818:
815:
814:
810:
789:Poisson process
785:Brownian motion
768:
762:
727:Maurice Fréchet
689:, and proved a
670:Poisson process
662:
646:natural numbers
629:
534:
502:Markov property
494:
478:
464:The adjectives
396:continuous-time
356:
204:Random variable
155:Bernoulli trial
28:
23:
22:
15:
12:
11:
5:
15934:
15924:
15923:
15918:
15913:
15908:
15891:
15890:
15888:
15887:
15882:
15880:List of topics
15876:
15873:
15872:
15870:
15869:
15864:
15859:
15854:
15849:
15844:
15839:
15837:Renewal theory
15834:
15829:
15824:
15819:
15814:
15809:
15804:
15802:Ergodic theory
15799:
15794:
15792:Control theory
15789:
15783:
15781:
15777:
15776:
15774:
15773:
15772:
15771:
15766:
15756:
15751:
15746:
15741:
15736:
15735:
15734:
15724:
15722:Snell envelope
15719:
15714:
15709:
15704:
15699:
15694:
15689:
15684:
15679:
15674:
15669:
15664:
15659:
15654:
15649:
15644:
15639:
15634:
15629:
15624:
15619:
15614:
15609:
15604:
15599:
15594:
15588:
15586:
15582:
15581:
15579:
15578:
15573:
15568:
15563:
15558:
15552:
15550:
15544:
15543:
15541:
15540:
15521:BorelâCantelli
15510:
15505:
15500:
15495:
15490:
15485:
15480:
15475:
15470:
15465:
15459:
15457:
15456:Limit theorems
15453:
15452:
15450:
15449:
15444:
15439:
15434:
15429:
15424:
15419:
15414:
15409:
15404:
15399:
15394:
15389:
15384:
15379:
15373:
15371:
15367:
15366:
15364:
15363:
15358:
15353:
15348:
15343:
15338:
15332:
15330:
15324:
15323:
15321:
15320:
15315:
15310:
15305:
15299:
15297:
15291:
15290:
15288:
15287:
15282:
15277:
15272:
15267:
15262:
15257:
15252:
15247:
15242:
15237:
15232:
15227:
15222:
15217:
15212:
15207:
15202:
15197:
15191:
15189:
15183:
15182:
15180:
15179:
15174:
15169:
15164:
15159:
15154:
15148:
15146:
15140:
15139:
15137:
15136:
15131:
15126:
15125:
15124:
15119:
15109:
15104:
15099:
15094:
15093:
15092:
15087:
15077:
15075:Hopfield model
15072:
15067:
15062:
15056:
15054:
15050:
15049:
15047:
15046:
15041:
15036:
15031:
15026:
15021:
15020:
15019:
15014:
15009:
15004:
14994:
14992:Markov process
14989:
14984:
14979:
14973:
14971:
14967:
14966:
14964:
14963:
14961:Wiener sausage
14958:
14956:Wiener process
14953:
14948:
14943:
14938:
14936:Stable process
14933:
14928:
14926:Semimartingale
14923:
14918:
14917:
14916:
14911:
14901:
14896:
14891:
14886:
14881:
14876:
14871:
14869:Jump diffusion
14866:
14861:
14856:
14851:
14846:
14844:Hawkes process
14841:
14836:
14831:
14826:
14824:Feller process
14821:
14816:
14811:
14806:
14801:
14796:
14791:
14789:Cauchy process
14786:
14785:
14784:
14779:
14774:
14769:
14764:
14754:
14753:
14752:
14742:
14740:Bessel process
14737:
14731:
14729:
14723:
14722:
14720:
14719:
14718:
14717:
14712:
14707:
14702:
14692:
14687:
14682:
14677:
14672:
14667:
14662:
14656:
14654:
14648:
14647:
14640:
14639:
14632:
14625:
14617:
14610:
14609:
14604:
14599:
14588:
14576:
14562:"Markov chain"
14557:
14555:
14554:External links
14552:
14550:
14549:
14534:
14519:
14512:
14494:
14480:
14470:
14452:John G. Kemeny
14449:
14431:
14421:
14403:
14382:
14353:
14317:
14298:
14280:
14262:
14240:
14233:
14225:
14223:
14220:
14217:
14216:
14178:
14149:
14111:
14075:
14049:
14042:
14020:
14005:Kenner, Hugh;
13997:
13979:
13958:
13932:
13908:
13890:
13870:
13863:
13845:
13818:
13749:
13721:
13718:on 2008-12-28.
13692:
13655:
13622:
13577:
13557:
13520:
13501:
13466:
13460:10.1.1.31.1768
13441:
13427:10.1.1.58.8652
13389:
13382:
13362:
13349:
13329:
13315:"Markov chain"
13301:
13244:
13218:
13179:
13144:
13125:(2): 101â112.
13109:
13074:
13033:
12998:
12979:(3): 269â279.
12963:
12906:
12847:
12803:
12768:
12733:
12686:(3): 950â976.
12666:
12660:
12634:
12627:
12599:
12570:(1): 143â154.
12547:
12523:
12498:
12491:
12462:
12440:
12433:
12415:
12402:(1): 295â297.
12382:
12375:
12355:
12330:
12323:
12303:
12286:(1 Dec 2023).
12284:Shalizi, Cosma
12272:
12265:
12247:
12219:
12191:
12184:
12166:
12153:
12130:
12117:(2): 246â290.
12095:
12052:
12027:(3): 387â405.
12007:
12001:10.1.1.28.6191
11982:
11959:
11952:
11923:
11916:
11887:
11880:
11860:
11843:
11836:
11816:
11809:
11789:
11783:978-0471667193
11782:
11756:
11749:
11726:
11719:
11696:
11668:
11661:
11641:
11622:(5): 395â398.
11606:
11599:
11579:
11572:
11552:
11539:(4): 509â546.
11516:
11483:
11476:
11446:
11427:(3): 291â292.
11411:
11392:(3): 255â257.
11373:
11362:(2): 253â268.
11346:
11339:
11322:10.1.1.114.632
11303:
11296:
11273:
11241:
11234:
11209:
11202:
11169:
11156:
11136:
11115:
11105:, Holden-Day.
11094:
11074:
11067:
11047:
11040:
11020:
11013:
10993:
10986:
10966:
10959:
10936:
10921:
10903:
10869:
10862:
10842:
10835:
10815:
10808:
10784:
10783:
10781:
10778:
10776:
10775:
10770:
10765:
10760:
10755:
10750:
10745:
10740:
10735:
10730:
10725:
10720:
10715:
10710:
10705:
10700:
10695:
10689:
10687:
10684:
10671:
10668:
10664:Mark V. Shaney
10647:
10644:
10626:
10623:
10598:
10597:
10594:
10591:
10588:
10584:
10583:
10580:
10577:
10574:
10570:
10569:
10566:
10563:
10560:
10556:
10555:
10552:
10549:
10546:
10542:
10541:
10538:
10535:
10532:
10528:
10527:
10524:
10521:
10518:
10514:
10513:
10510:
10507:
10504:
10500:
10499:
10496:
10493:
10490:
10486:
10485:
10482:
10479:
10476:
10472:
10471:
10468:
10465:
10462:
10451:
10450:
10447:
10444:
10441:
10432:
10431:
10428:
10425:
10422:
10413:
10412:
10409:
10406:
10403:
10399:
10398:
10390:
10382:
10379:
10333:
10330:
10317:
10314:
10274:path-dependent
10269:
10266:
10220:
10217:
10200:
10197:
10180:
10158:
10154:
10151:
10148:
10123:
10119:
10116:
10113:
10107:
10100:
10096:
10092:
10068:
10064:
10043:
10023:
10003:
9969:
9965:
9962:
9959:
9953:
9946:
9942:
9938:
9920:
9917:
9862:Main article:
9859:
9856:
9841:bioinformatics
9791:Claude Shannon
9786:
9783:
9771:
9768:
9755:
9752:
9748:Markov blanket
9743:
9740:
9739:
9738:
9732:
9726:
9720:
9710:
9695:bioinformatics
9683:
9680:
9669:steric effects
9625:
9609:
9592:
9563:
9556:
9545:
9538:
9523:
9508:
9507:
9506:
9504:
9501:
9486:thermodynamics
9481:
9478:
9473:
9470:
9410:Main article:
9407:
9404:
9373:Main article:
9370:
9367:
9364:
9363:
9358:
9353:
9349:
9348:
9343:
9340:
9336:
9335:
9332:
9329:
9316:Main article:
9313:
9310:
9308:
9305:
9262:
9261:
9250:
9243:
9238:
9232:
9229:
9225:
9221:
9218:
9215:
9212:
9209:
9206:
9203:
9199:
9191:
9188:
9184:
9180:
9177:
9174:
9171:
9168:
9165:
9162:
9159:
9153:
9150:
9120:
9116:
9112:
9109:
9088:
9068:
9057:
9056:
9045:
9042:
9039:
9036:
9033:
9010:
8972:
8971:
8960:
8955:
8952:
8947:
8943:
8940:
8937:
8934:
8931:
8927:
8922:
8919:
8916:
8913:
8895:
8894:
8881:
8876:
8868:
8866:
8863:
8862:
8859:
8856:
8853:
8845:
8838:
8835:
8831:
8825:
8822:
8819:
8815:
8808:
8805:
8801:
8795:
8794:
8792:
8787:
8782:
8779:
8775:
8745:
8708:
8705:
8675:
8672:
8669:
8665:
8661:
8656:
8649:
8646:
8628:
8621:
8618:
8617:
8616:
8601:
8598:
8595:
8592:
8584:
8582:
8579:
8574:
8569:
8565:
8559:
8556:
8552:
8546:
8543:
8540:
8536:
8532:
8529:
8528:
8525:
8522:
8519:
8511:
8509:
8506:
8501:
8496:
8492:
8488:
8487:
8465:expected value
8450:
8445:
8441:
8416:
8413:
8400:Main article:
8397:
8394:
8387:autoregressive
8372:
8371:
8360:
8355:
8349:
8346:
8343:
8340:
8337:
8334:
8331:
8328:
8325:
8322:
8319:
8316:
8313:
8310:
8307:
8304:
8301:
8298:
8293:
8288:
8285:
8282:
8279:
8276:
8245:
8242:
8218:
8198:
8195:
8192:
8172:
8169:
8166:
8163:
8160:
8155:
8152:
8148:
8127:
8116:ergodic theory
8090:
8085:
8081:
8078:
8058:
8055:
8052:
8047:
8043:
8039:
8036:
8033:
8030:
8027:
8022:
8018:
8014:
8009:
8005:
8001:
7998:
7978:
7975:
7972:
7969:
7966:
7945:
7922:
7917:
7913:
7910:
7894:
7891:
7890:
7889:
7873:
7853:
7850:
7847:
7844:
7841:
7838:
7835:
7832:
7829:
7826:
7823:
7820:
7817:
7814:
7811:
7787:
7767:
7764:
7761:
7758:
7755:
7752:
7749:
7746:
7743:
7732:
7720:
7717:
7714:
7711:
7708:
7686:
7682:
7661:
7658:
7655:
7651:
7648:
7645:
7642:
7630:
7618:
7615:
7612:
7609:
7606:
7603:
7600:
7580:
7577:
7574:
7554:
7543:
7531:
7523:
7520:
7517:
7514:
7511:
7508:
7505:
7502:
7499:
7479:
7459:
7456:
7451:
7447:
7443:
7440:
7437:
7434:
7431:
7407:
7384:
7381:
7378:
7374:
7371:
7368:
7365:
7344:
7322:
7318:
7297:
7277:
7274:
7255:
7251:
7230:
7212:ergodic theory
7203:
7200:
7180:are positive.
7167:
7163:
7142:
7122:is said to be
7115:
7112:
7099:
7094:
7090:
7086:
7083:
7079:
7075:
7072:
7067:
7063:
7047:
7046:Irreducibility
7044:
7028:null recurrent
7026:is finite and
7013:
7009:
6989:
6988:
6977:
6972:
6969:
6966:
6961:
6958:
6954:
6950:
6947:
6942:
6937:
6934:
6931:
6927:
6923:
6920:
6915:
6911:
6907:
6904:
6901:
6896:
6892:
6851:is said to be
6829:
6826:
6823:
6803:
6800:
6797:
6782:
6781:
6770:
6767:
6764:
6761:
6758:
6755:
6750:
6746:
6742:
6739:
6736:
6731:
6727:
6723:
6720:
6717:
6714:
6711:
6708:
6705:
6702:
6699:
6696:
6631:
6628:
6607:
6604:
6592:
6589:
6584:Main article:
6581:
6578:
6567:
6560:
6545:
6541:
6535:
6531:
6526:
6522:
6519:
6516:
6512:
6506:
6502:
6497:
6484:
6477:
6460:
6454:
6450:approaches to
6435:
6423:
6422:
6406:
6399:
6393:
6388:
6381:
6376:
6369:
6365:
6359:
6355:
6349:
6342:
6338:
6334:
6331:
6328:
6322:
6316:
6311:
6304:
6299:
6292:
6288:
6282:
6278:
6272:
6265:
6261:
6257:
6251:
6245:
6240:
6233:
6228:
6221:
6217:
6211:
6207:
6201:
6194:
6190:
6186:
6180:
6174:
6169:
6162:
6158:
6153:
6147:
6142:
6138:
6134:
6131:
6129:
6127:
6124:
6121:
6118:
6108:
6104:
6100:
6095:
6091:
6087:
6081:
6075:
6070:
6063:
6058:
6054:
6048:
6044:
6040:
6037:
6034:
6028:
6022:
6017:
6010:
6005:
6001:
5995:
5991:
5987:
5981:
5975:
5970:
5963:
5958:
5954:
5948:
5944:
5940:
5937:
5935:
5933:
5928:
5925:
5920:
5913:
5908:
5905:
5899:
5892:
5886:
5881:
5874:
5870:
5866:
5863:
5860:
5854:
5848:
5843:
5836:
5832:
5828:
5822:
5816:
5811:
5804:
5800:
5795:
5791:
5788:
5786:
5784:
5779:
5776:
5771:
5764:
5759:
5756:
5753:
5748:
5745:
5743:
5741:
5737:
5732:
5729:
5724:
5721:
5718:
5713:
5709:
5705:
5700:
5697:
5692:
5689:
5686:
5681:
5676:
5671:
5668:
5663:
5660:
5657:
5652:
5647:
5643:
5640:
5638:
5634:
5631:
5628:
5623:
5618:
5617:
5586:
5580:
5554:
5553:
5542:
5538:
5534:
5529:
5525:
5520:
5515:
5510:
5503:
5499:
5493:
5488:
5485:
5482:
5478:
5474:
5468:
5462:
5437:
5432:
5427:
5411:
5394:
5384:
5376:-th column of
5367:
5350:
5349:
5338:
5334:
5328:
5324:
5319:
5315:
5312:
5309:
5305:
5299:
5295:
5290:
5286:
5282:
5276:
5272:
5267:
5263:
5259:
5253:
5249:
5244:
5240:
5237:
5223:
5222:
5211:
5206:
5203:
5198:
5195:
5192:
5187:
5183:
5162:
5155:
5148:
5141:
5066:
5062:
5057:
5053:
5049:
5036:
5033:
5002:
4992:
4991:
4979:
4968:
4963:
4960:
4956:
4952:
4947:
4942:
4937:
4933:
4929:
4926:
4923:
4920:
4915:
4912:
4909:
4904:
4899:
4896:
4893:
4889:
4800:
4783:
4777:
4776:
4765:
4760:
4757:
4754:
4749:
4744:
4741:
4736:
4731:
4726:
4722:
4718:
4714:
4695:
4694:
4683:
4679:
4675:
4671:
4668:
4642:
4637:
4632:
4625:
4622:
4619:
4615:
4611:
4607:
4579:
4578:
4565:
4557:
4554:
4549:
4545:
4542:
4537:
4536:
4534:
4529:
4524:
4518:
4515:
4513:
4510:
4509:
4506:
4503:
4501:
4498:
4497:
4495:
4488:
4480:
4477:
4472:
4468:
4465:
4460:
4459:
4457:
4445:
4433:
4429:
4424:
4421:
4418:
4415:
4410:
4404:
4401:
4396:
4393:
4388:
4380:
4374:
4371:
4369:
4366:
4365:
4362:
4359:
4357:
4354:
4353:
4351:
4346:
4342:
4316:
4311:
4304:
4301:
4298:
4294:
4264:
4259:
4252:
4249:
4246:
4242:
4222:
4221:
4210:
4206:
4202:
4197:
4192:
4185:
4182:
4179:
4175:
4128:
4125:
4104:
4101:
4096:
4092:
4088:
4085:
4080:
4076:
4048:
4044:
4006:
4003:
3998:
3994:
3988:
3984:
3972:
3971:
3954:
3950:
3943:
3939:
3934:
3929:
3926:
3908:
3907:
3896:
3893:
3890:
3886:
3882:
3859:
3856:
3840:
3839:
3828:
3825:
3822:
3819:
3814:
3810:
3806:
3803:
3800:
3795:
3792:
3789:
3785:
3781:
3778:
3775:
3770:
3767:
3763:
3723:
3720:
3712:
3711:
3700:
3697:
3694:
3691:
3688:
3685:
3682:
3679:
3676:
3672:
3669:
3643:
3637:
3636:
3625:
3620:
3616:
3612:
3607:
3604:
3601:
3597:
3593:
3586:
3583:
3580:
3576:
3570:
3566:
3561:
3557:
3554:
3549:
3545:
3541:
3534:
3530:
3525:
3521:
3518:
3515:
3510:
3506:
3502:
3495:
3491:
3486:
3482:
3477:
3473:
3469:
3462:
3458:
3453:
3449:
3444:
3441:
3438:
3434:
3430:
3423:
3420:
3417:
3413:
3408:
3404:
3401:
3386:
3379:
3372:
3365:
3358:
3351:
3344:
3332:For any value
3329:
3326:
3318:
3310:
3306:
3293:
3286:
3279:
3272:
3259:
3252:
3249:
3226:
3223:
3219:
3188:
3185:
3181:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3137:
3134:
3130:
3126:
3121:
3118:
3114:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3056:
3023:
3019:
3016:
3013:
3010:
3005:
3001:
2996:
2975:
2972:
2967:
2964:
2961:
2957:
2936:
2933:
2928:
2924:
2889:
2885:
2864:
2861:
2851:
2834:
2809: â„ 0
2805:
2799:
2789:Main article:
2786:
2783:
2782:
2781:
2768:
2762:
2759:
2756:
2753:
2750:
2746:
2742:
2739:
2736:
2731:
2728:
2725:
2721:
2717:
2712:
2708:
2703:
2699:
2694:
2690:
2680:values, i.e.,
2661:
2656:
2652:
2648:
2628:
2623:
2619:
2615:
2587:
2584:
2581:
2573:
2568:
2565:
2562:
2558:
2554:
2549:
2546:
2543:
2539:
2535:
2532:
2529:
2524:
2521:
2518:
2514:
2510:
2505:
2502:
2499:
2495:
2491:
2486:
2483:
2480:
2476:
2472:
2467:
2464:
2461:
2457:
2453:
2448:
2444:
2440:
2435:
2431:
2427:
2424:
2421:
2419:
2417:
2414:
2413:
2410:
2405:
2401:
2397:
2392:
2388:
2384:
2381:
2378:
2373:
2370:
2367:
2363:
2359:
2354:
2351:
2348:
2344:
2340:
2335:
2332:
2329:
2325:
2321:
2316:
2313:
2310:
2306:
2302:
2297:
2293:
2289:
2284:
2280:
2276:
2273:
2270:
2268:
2264:
2263:
2244:
2230:
2226:
2197:
2192:
2188:
2184:
2179:
2176:
2173:
2169:
2165:
2162:
2159:
2154:
2150:
2146:
2141:
2138:
2135:
2131:
2127:
2122:
2118:
2114:
2109:
2105:
2101:
2098:
2095:
2092:
2087:
2083:
2079:
2074:
2070:
2066:
2063:
2060:
2055:
2051:
2047:
2042:
2038:
2034:
2029:
2025:
2021:
2016:
2012:
2008:
2005:
1994:
1974:
1971:
1968:
1963:
1960:
1957:
1953:
1949:
1946:
1943:
1938:
1934:
1930:
1927:
1924:
1921:
1918:
1915:
1910:
1906:
1902:
1899:
1896:
1891:
1888:
1885:
1881:
1877:
1874:
1861:
1858:
1844:
1838:
1837:
1826:
1823:
1820:
1815:
1811:
1807:
1802:
1798:
1794:
1791:
1788:
1783:
1779:
1775:
1770:
1766:
1762:
1759:
1735:
1732:
1727:
1723:
1719:
1714:
1710:
1706:
1703:
1700:
1695:
1692:
1689:
1685:
1681:
1678:
1675:
1672:
1667:
1663:
1659:
1654:
1650:
1646:
1643:
1640:
1635:
1631:
1627:
1622:
1618:
1614:
1609:
1605:
1601:
1596:
1592:
1588:
1585:
1582:
1577:
1574:
1571:
1567:
1563:
1560:
1541:
1534:
1527:
1514:Main article:
1511:
1508:
1506:
1503:
1490:
1487:
1484:
1481:
1478:
1475:
1470:
1467:
1464:
1460:
1439:
1436:
1433:
1430:
1427:
1424:
1419:
1415:
1393:
1390:
1387:
1384:
1381:
1378:
1373:
1369:
1346:
1342:
1319:
1315:
1294:
1291:
1288:
1285:
1282:
1279:
1274:
1270:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1210:
1207:
1204:
1201:
1198:
1195:
1190:
1186:
1159:
1155:
1128:
1124:
1098:
1094:
1071:
1067:
1044:
1040:
1019:
1016:
1013:
1008:
1004:
981:
977:
956:
953:
950:
945:
941:
913:
909:
905:
902:
899:
894:
890:
886:
866:
863:
858:
854:
827:
823:
809:
806:
805:
804:
800:
792:
781:Wiener process
776:gambler's ruin
764:Main article:
761:
758:
750:William Feller
742:Sydney Chapman
738:Norbert Wiener
714:Francis Galton
698:Henri Poincaré
674:Pavel Nekrasov
661:
658:
628:
625:
591:
590:
587:Wiener process
579:
576:
572:
571:
566:(for example,
561:
558:
557:Discrete-time
554:
553:
550:
547:
533:
530:
506:memorylessness
493:
490:
477:
474:
406:mathematician
368:Markov process
358:
357:
355:
354:
347:
340:
332:
329:
328:
327:
326:
321:
313:
312:
311:
310:
305:
303:Bayes' theorem
300:
295:
290:
285:
277:
276:
275:
274:
269:
264:
259:
251:
250:
249:
248:
247:
246:
241:
236:
234:Observed value
231:
226:
221:
219:Expected value
216:
211:
201:
196:
195:
194:
189:
184:
179:
174:
169:
159:
158:
157:
147:
146:
145:
140:
135:
130:
125:
115:
110:
102:
101:
100:
99:
94:
89:
88:
87:
77:
76:
75:
62:
61:
53:
52:
46:
45:
26:
9:
6:
4:
3:
2:
15933:
15922:
15919:
15917:
15914:
15912:
15911:Markov models
15909:
15907:
15904:
15903:
15901:
15886:
15883:
15881:
15878:
15877:
15874:
15868:
15865:
15863:
15860:
15858:
15855:
15853:
15850:
15848:
15845:
15843:
15840:
15838:
15835:
15833:
15830:
15828:
15825:
15823:
15820:
15818:
15815:
15813:
15810:
15808:
15805:
15803:
15800:
15798:
15795:
15793:
15790:
15788:
15785:
15784:
15782:
15778:
15770:
15767:
15765:
15762:
15761:
15760:
15757:
15755:
15752:
15750:
15747:
15745:
15742:
15740:
15739:Stopping time
15737:
15733:
15730:
15729:
15728:
15725:
15723:
15720:
15718:
15715:
15713:
15710:
15708:
15705:
15703:
15700:
15698:
15695:
15693:
15690:
15688:
15685:
15683:
15680:
15678:
15675:
15673:
15670:
15668:
15665:
15663:
15660:
15658:
15655:
15653:
15650:
15648:
15645:
15643:
15640:
15638:
15635:
15633:
15630:
15628:
15625:
15623:
15620:
15618:
15615:
15613:
15610:
15608:
15605:
15603:
15600:
15598:
15595:
15593:
15590:
15589:
15587:
15583:
15577:
15574:
15572:
15569:
15567:
15564:
15562:
15559:
15557:
15554:
15553:
15551:
15549:
15545:
15538:
15534:
15530:
15529:HewittâSavage
15526:
15522:
15518:
15514:
15513:Zeroâone laws
15511:
15509:
15506:
15504:
15501:
15499:
15496:
15494:
15491:
15489:
15486:
15484:
15481:
15479:
15476:
15474:
15471:
15469:
15466:
15464:
15461:
15460:
15458:
15454:
15448:
15445:
15443:
15440:
15438:
15435:
15433:
15430:
15428:
15425:
15423:
15420:
15418:
15415:
15413:
15410:
15408:
15405:
15403:
15400:
15398:
15395:
15393:
15390:
15388:
15385:
15383:
15380:
15378:
15375:
15374:
15372:
15368:
15362:
15359:
15357:
15354:
15352:
15349:
15347:
15344:
15342:
15339:
15337:
15334:
15333:
15331:
15329:
15325:
15319:
15316:
15314:
15311:
15309:
15306:
15304:
15301:
15300:
15298:
15296:
15292:
15286:
15283:
15281:
15278:
15276:
15273:
15271:
15268:
15266:
15263:
15261:
15258:
15256:
15253:
15251:
15248:
15246:
15243:
15241:
15238:
15236:
15233:
15231:
15228:
15226:
15223:
15221:
15218:
15216:
15213:
15211:
15210:BlackâScholes
15208:
15206:
15203:
15201:
15198:
15196:
15193:
15192:
15190:
15188:
15184:
15178:
15175:
15173:
15170:
15168:
15165:
15163:
15160:
15158:
15155:
15153:
15150:
15149:
15147:
15145:
15141:
15135:
15132:
15130:
15127:
15123:
15120:
15118:
15115:
15114:
15113:
15112:Point process
15110:
15108:
15105:
15103:
15100:
15098:
15095:
15091:
15088:
15086:
15083:
15082:
15081:
15078:
15076:
15073:
15071:
15070:Gibbs measure
15068:
15066:
15063:
15061:
15058:
15057:
15055:
15051:
15045:
15042:
15040:
15037:
15035:
15032:
15030:
15027:
15025:
15022:
15018:
15015:
15013:
15010:
15008:
15005:
15003:
15000:
14999:
14998:
14995:
14993:
14990:
14988:
14985:
14983:
14980:
14978:
14975:
14974:
14972:
14968:
14962:
14959:
14957:
14954:
14952:
14949:
14947:
14944:
14942:
14939:
14937:
14934:
14932:
14929:
14927:
14924:
14922:
14919:
14915:
14912:
14910:
14907:
14906:
14905:
14902:
14900:
14897:
14895:
14892:
14890:
14887:
14885:
14882:
14880:
14877:
14875:
14872:
14870:
14867:
14865:
14862:
14860:
14859:ItĂŽ diffusion
14857:
14855:
14852:
14850:
14847:
14845:
14842:
14840:
14837:
14835:
14834:Gamma process
14832:
14830:
14827:
14825:
14822:
14820:
14817:
14815:
14812:
14810:
14807:
14805:
14802:
14800:
14797:
14795:
14792:
14790:
14787:
14783:
14780:
14778:
14775:
14773:
14770:
14768:
14765:
14763:
14760:
14759:
14758:
14755:
14751:
14748:
14747:
14746:
14743:
14741:
14738:
14736:
14733:
14732:
14730:
14728:
14724:
14716:
14713:
14711:
14708:
14706:
14705:Self-avoiding
14703:
14701:
14698:
14697:
14696:
14693:
14691:
14690:Moran process
14688:
14686:
14683:
14681:
14678:
14676:
14673:
14671:
14668:
14666:
14663:
14661:
14658:
14657:
14655:
14653:
14652:Discrete time
14649:
14645:
14638:
14633:
14631:
14626:
14624:
14619:
14618:
14615:
14608:
14605:
14603:
14600:
14598:
14594:
14589:
14587:
14583:
14580:
14577:
14573:
14569:
14568:
14563:
14559:
14558:
14547:
14543:
14539:
14535:
14532:
14531:0-7923-9650-2
14528:
14524:
14520:
14517:
14513:
14510:
14509:0-471-33341-7
14506:
14502:
14498:
14495:
14493:
14489:
14485:
14481:
14479:
14478:0-521-60494-X
14475:
14471:
14469:
14468:0-442-04328-7
14465:
14461:
14457:
14453:
14450:
14447:
14440:
14439:
14432:
14427:
14422:
14420:
14416:
14412:
14408:
14404:
14401:
14397:
14385:
14379:
14375:
14371:
14366:
14365:
14356:
14350:
14346:
14342:
14338:
14334:
14329:
14328:
14322:
14318:
14315:
14311:
14310:0-387-19832-6
14307:
14303:
14299:
14296:
14295:0-471-52369-0
14292:
14288:
14284:
14281:
14278:
14277:0-89871-296-3
14274:
14271:
14267:
14263:
14259:
14255:
14251:
14247:
14241:
14238:
14234:
14231:
14227:
14226:
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14200:
14196:
14192:
14185:
14183:
14167:
14160:
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14138:
14134:
14130:
14126:
14122:
14115:
14106:
14101:
14097:
14093:
14089:
14082:
14080:
14071:
14067:
14063:
14056:
14054:
14045:
14039:
14034:
14033:
14024:
14016:
14012:
14008:
14001:
13993:
13989:
13983:
13969:
13962:
13948:on 2007-12-09
13947:
13943:
13936:
13929:
13925:
13921:
13918:
13912:
13904:
13900:
13899:"Continuator"
13894:
13888:
13884:
13880:
13874:
13866:
13860:
13857:. MIT Press.
13856:
13849:
13841:
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13822:
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13810:
13805:
13800:
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13776:
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13753:
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13725:
13714:
13710:
13703:
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13684:
13679:
13674:
13670:
13666:
13659:
13651:
13647:
13643:
13639:
13638:
13633:
13626:
13618:
13614:
13610:
13606:
13601:
13596:
13593:(2): 357â84.
13592:
13588:
13581:
13573:
13569:
13561:
13552:
13547:
13543:
13539:
13535:
13531:
13524:
13516:
13512:
13511:Am. Econ. Rev
13505:
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13097:
13093:
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13078:
13069:
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13048:
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13025:
13021:
13017:
13013:
13009:
13002:
12994:
12990:
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12978:
12974:
12967:
12959:
12955:
12950:
12945:
12941:
12937:
12933:
12929:
12925:
12921:
12920:AIChE Journal
12917:
12910:
12902:
12898:
12893:
12888:
12883:
12878:
12874:
12870:
12866:
12862:
12858:
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12843:
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12807:
12799:
12795:
12791:
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12783:
12779:
12772:
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12756:
12752:
12748:
12744:
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12725:
12720:
12715:
12711:
12707:
12703:
12699:
12694:
12689:
12685:
12681:
12677:
12670:
12663:
12661:9781441967657
12657:
12653:
12649:
12645:
12638:
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12616:
12612:
12611:
12603:
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12591:
12586:
12581:
12577:
12573:
12569:
12565:
12561:
12554:
12552:
12534:
12527:
12521:
12517:
12513:
12507:
12505:
12503:
12494:
12492:9780511810633
12488:
12484:
12480:
12476:
12475:Markov Chains
12472:
12471:Norris, J. R.
12466:
12458:
12451:
12444:
12436:
12430:
12426:
12419:
12410:
12405:
12401:
12397:
12393:
12386:
12378:
12372:
12368:
12367:
12359:
12344:
12340:
12334:
12326:
12320:
12316:
12315:
12307:
12293:
12289:
12285:
12279:
12277:
12268:
12266:0-8162-6664-6
12262:
12258:
12251:
12233:
12229:
12228:Lalley, Steve
12223:
12209:
12205:
12201:
12195:
12187:
12185:0-07-028631-0
12181:
12177:
12170:
12156:
12154:9780719022067
12150:
12146:
12145:
12140:
12134:
12125:
12120:
12116:
12112:
12111:
12106:
12099:
12091:
12087:
12083:
12079:
12075:
12071:
12068:(4): 041112.
12067:
12063:
12056:
12042:
12038:
12034:
12030:
12026:
12022:
12018:
12011:
12002:
11997:
11993:
11986:
11969:
11963:
11955:
11949:
11945:
11941:
11937:
11930:
11928:
11919:
11917:9780511810633
11913:
11909:
11905:
11901:
11900:Markov Chains
11897:
11896:Norris, J. R.
11891:
11883:
11877:
11873:
11872:
11864:
11856:
11855:
11847:
11839:
11833:
11829:
11828:
11820:
11812:
11806:
11802:
11801:
11793:
11785:
11779:
11775:
11771:
11768:. p. 1.
11767:
11760:
11752:
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11741:
11733:
11731:
11722:
11716:
11712:
11711:
11703:
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11672:
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11645:
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11621:
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11602:
11596:
11592:
11591:
11583:
11575:
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11556:
11547:
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11530:
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11508:
11504:
11500:
11492:
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11473:
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11199:
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11178:
11176:
11174:
11166:
11160:
11154:
11153:0-19-920613-9
11150:
11146:
11140:
11133:
11132:0-19-920613-9
11129:
11125:
11119:
11112:
11111:0-8162-6664-6
11108:
11104:
11098:
11092:
11091:0-521-81099-X
11088:
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10997:
10989:
10983:
10979:
10978:
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10956:
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10943:
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10932:
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10924:
10918:
10914:
10907:
10899:
10891:
10887:
10886:
10880:
10873:
10865:
10859:
10856:. CRC Press.
10855:
10854:
10846:
10838:
10832:
10828:
10827:
10819:
10811:
10805:
10801:
10800:
10792:
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10709:
10706:
10704:
10701:
10699:
10696:
10694:
10691:
10690:
10683:
10681:
10677:
10667:
10665:
10661:
10657:
10653:
10643:
10641:
10637:
10636:base stealing
10633:
10622:
10618:
10615:
10613:
10609:
10605:
10595:
10592:
10589:
10586:
10585:
10581:
10578:
10575:
10572:
10571:
10567:
10564:
10561:
10558:
10557:
10553:
10550:
10547:
10544:
10543:
10539:
10536:
10533:
10530:
10529:
10525:
10522:
10519:
10516:
10515:
10511:
10508:
10505:
10502:
10501:
10497:
10494:
10491:
10488:
10487:
10483:
10480:
10477:
10474:
10473:
10469:
10466:
10463:
10460:
10459:
10448:
10445:
10442:
10434:
10433:
10429:
10426:
10423:
10415:
10414:
10410:
10407:
10404:
10401:
10400:
10391:
10383:
10380:
10377:
10376:
10370:
10368:
10364:
10359:
10355:
10354:SuperCollider
10351:
10347:
10343:
10339:
10329:
10327:
10323:
10313:
10311:
10307:
10306:authoritarian
10303:
10299:
10295:
10291:
10286:
10285:
10279:
10275:
10265:
10263:
10259:
10257:
10252:
10250:
10246:
10242:
10238:
10234:
10230:
10226:
10216:
10214:
10210:
10206:
10196:
10192:
10178:
10156:
10152:
10149:
10146:
10121:
10117:
10114:
10111:
10105:
10098:
10094:
10090:
10066:
10062:
10041:
10021:
10001:
9993:
9989:
9967:
9963:
9960:
9957:
9951:
9944:
9940:
9936:
9925:
9916:
9914:
9910:
9906:
9902:
9898:
9894:
9890:
9886:
9882:
9877:
9875:
9871:
9865:
9855:
9853:
9849:
9844:
9842:
9838:
9833:
9831:
9827:
9823:
9819:
9815:
9811:
9806:
9802:
9798:
9797:
9792:
9782:
9780:
9776:
9767:
9764:
9760:
9751:
9749:
9736:
9733:
9730:
9727:
9724:
9721:
9718:
9714:
9711:
9708:
9704:
9700:
9697:, where most
9696:
9692:
9691:Phylogenetics
9689:
9688:
9687:
9679:
9677:
9672:
9670:
9665:
9660:
9657:
9652:
9650:
9645:
9643:
9638:
9628:
9607:
9554:
9543:
9521:
9500:
9499:simulations.
9498:
9493:
9491:
9487:
9477:
9469:
9467:
9463:
9459:
9458:sofic systems
9455:
9454:Chacon system
9451:
9447:
9443:
9439:
9435:
9431:
9427:
9423:
9419:
9413:
9403:
9401:
9398:
9394:
9389:
9387:
9382:
9376:
9362:
9359:
9357:
9354:
9351:
9350:
9347:
9344:
9342:Markov chain
9341:
9338:
9337:
9333:
9330:
9328:
9327:
9324:
9319:
9304:
9302:
9298:
9294:
9290:
9286:
9281:
9279:
9272:is not. Once
9271:
9267:
9248:
9241:
9230:
9227:
9216:
9210:
9207:
9201:
9189:
9186:
9175:
9169:
9166:
9160:
9157:
9151:
9148:
9141:
9140:
9139:
9133:
9118:
9110:
9086:
9066:
9043:
9040:
9037:
9034:
9031:
9024:
9023:
9022:
9008:
8999:
8997:
8993:
8992:main diagonal
8989:
8985:
8981:
8977:
8958:
8953:
8950:
8945:
8938:
8932:
8929:
8925:
8920:
8917:
8914:
8911:
8904:
8903:
8902:
8900:
8874:
8864:
8857:
8854:
8851:
8836:
8833:
8829:
8823:
8820:
8817:
8813:
8806:
8803:
8799:
8790:
8785:
8780:
8777:
8773:
8765:
8764:
8763:
8761:
8757:
8753:
8748:
8744:
8740:
8736:
8735:
8730:
8726:
8722:
8714:
8704:
8702:
8698:
8693:
8691:
8690:Kelly's lemma
8673:
8670:
8667:
8663:
8659:
8654:
8644:
8631:
8627:
8620:Time reversal
8599:
8596:
8593:
8590:
8580:
8577:
8572:
8567:
8563:
8557:
8554:
8550:
8544:
8541:
8538:
8534:
8530:
8523:
8520:
8517:
8507:
8504:
8499:
8494:
8490:
8478:
8477:
8476:
8474:
8470:
8466:
8448:
8443:
8439:
8430:
8427:, the vector
8426:
8423: â
8422:
8412:
8409:
8403:
8396:Hitting times
8393:
8391:
8388:
8383:
8381:
8377:
8358:
8341:
8335:
8332:
8326:
8320:
8314:
8311:
8308:
8302:
8296:
8286:
8280:
8274:
8267:
8266:
8265:
8263:
8259:
8255:
8251:
8241:
8239:
8235:
8230:
8193:
8190:
8170:
8167:
8161:
8153:
8150:
8146:
8125:
8117:
8112:
8110:
8105:
8079:
8053:
8050:
8045:
8041:
8034:
8028:
8025:
8020:
8016:
8012:
8007:
8003:
7996:
7967:
7964:
7911:
7900:
7887:
7871:
7848:
7845:
7842:
7839:
7836:
7830:
7827:
7821:
7818:
7815:
7809:
7801:
7785:
7762:
7759:
7756:
7750:
7747:
7744:
7741:
7733:
7718:
7715:
7712:
7709:
7706:
7684:
7680:
7656:
7631:
7616:
7613:
7610:
7607:
7604:
7601:
7598:
7578:
7575:
7572:
7552:
7544:
7529:
7521:
7515:
7509:
7503:
7497:
7477:
7457:
7454:
7449:
7441:
7438:
7435:
7429:
7421:
7420:
7419:
7405:
7396:
7379:
7342:
7320:
7316:
7295:
7287:
7283:
7273:
7271:
7253:
7249:
7228:
7220:
7215:
7213:
7209:
7199:
7196:
7194:
7190:
7186:
7181:
7165:
7161:
7140:
7131:
7129:
7125:
7121:
7111:
7092:
7088:
7081:
7077:
7073:
7070:
7065:
7061:
7051:
7043:
7041:
7037:
7032:
7029:
7011:
7007:
6998:
6994:
6975:
6967:
6959:
6956:
6952:
6948:
6945:
6935:
6932:
6929:
6925:
6921:
6913:
6909:
6902:
6899:
6894:
6890:
6882:
6881:
6880:
6878:
6874:
6870:
6866:
6862:
6858:
6854:
6850:
6845:
6843:
6827:
6824:
6821:
6801:
6798:
6795:
6787:
6784:The state is
6765:
6762:
6756:
6753:
6748:
6744:
6740:
6737:
6734:
6729:
6725:
6715:
6712:
6709:
6706:
6697:
6694:
6687:
6686:
6685:
6682:
6676:
6671:
6666:
6660:
6654:
6648:
6646:
6642:
6637:
6627:
6625:
6621:
6617:
6613:
6603:
6600:
6598:
6597:Harris chains
6591:Harris chains
6587:
6577:
6575:
6566:
6559:
6543:
6533:
6529:
6520:
6517:
6514:
6504:
6500:
6483:
6476:
6472:
6468:
6459:
6453:
6449:
6443:
6434:
6430:
6404:
6391:
6379:
6374:
6367:
6363:
6357:
6353:
6347:
6340:
6336:
6332:
6329:
6326:
6314:
6302:
6297:
6290:
6286:
6280:
6276:
6270:
6263:
6259:
6255:
6243:
6231:
6226:
6219:
6215:
6209:
6205:
6199:
6192:
6188:
6184:
6172:
6160:
6156:
6151:
6145:
6140:
6136:
6132:
6130:
6122:
6119:
6116:
6106:
6102:
6093:
6089:
6073:
6061:
6056:
6052:
6046:
6042:
6038:
6035:
6032:
6020:
6008:
6003:
5999:
5993:
5989:
5985:
5973:
5961:
5956:
5952:
5946:
5942:
5938:
5936:
5926:
5923:
5911:
5897:
5884:
5872:
5868:
5864:
5861:
5858:
5846:
5834:
5830:
5826:
5814:
5802:
5798:
5793:
5789:
5787:
5777:
5774:
5762:
5746:
5744:
5735:
5730:
5727:
5711:
5707:
5703:
5698:
5695:
5679:
5674:
5669:
5666:
5650:
5641:
5639:
5629:
5608:
5607:
5606:
5604:
5600:
5596:
5592:
5585:
5579:
5575:
5569:
5563:
5559:
5540:
5532:
5527:
5523:
5518:
5513:
5501:
5497:
5491:
5486:
5483:
5480:
5476:
5472:
5451:
5450:
5449:
5448:we can write
5435:
5430:
5414:
5410:
5406:
5402:
5397:
5392:
5387:
5383:
5379:
5375:
5370:
5366:
5362:
5355:
5336:
5326:
5322:
5313:
5310:
5307:
5297:
5293:
5284:
5274:
5270:
5261:
5251:
5247:
5238:
5235:
5228:
5227:
5226:
5209:
5204:
5201:
5185:
5173:
5172:
5171:
5170:
5165:
5161:
5154:
5147:
5140:
5136:
5132:
5128:
5124:
5120:
5115:
5113:
5109:
5105:
5101:
5097:
5093:
5089:
5086:
5082:
5064:
5051:
5032:
5030:
5026:
5022:
5019:. Hence, the
5018:
5014:
5009:
5005:
5001:
4997:
4988:
4984:
4980:
4966:
4961:
4958:
4945:
4935:
4924:
4913:
4910:
4907:
4894:
4891:
4879:
4878:
4877:
4875:
4871:
4867:
4862:
4860:
4856:
4852:
4848:
4844:
4840:
4836:
4832:
4828:
4824:
4820:
4816:
4812:
4807:
4803:
4799:
4795:
4791:
4786:
4782:
4763:
4758:
4755:
4752:
4742:
4734:
4724:
4704:
4703:
4702:
4700:
4681:
4673:
4658:
4657:
4656:
4653:
4640:
4635:
4617:
4609:
4596:
4592:
4588:
4582:
4563:
4555:
4552:
4543:
4540:
4532:
4527:
4522:
4516:
4511:
4504:
4499:
4493:
4486:
4478:
4475:
4466:
4463:
4455:
4446:
4427:
4422:
4419:
4416:
4413:
4402:
4399:
4394:
4391:
4378:
4372:
4367:
4360:
4355:
4349:
4344:
4332:
4331:
4330:
4314:
4296:
4283:
4278:
4262:
4244:
4231:
4227:
4208:
4200:
4195:
4177:
4165:
4164:
4163:
4157:
4148:
4146:
4142:
4138:
4134:
4124:
4122:
4118:
4102:
4099:
4094:
4090:
4086:
4083:
4078:
4074:
4065:
4046:
4042:
4031:
4028:
4024:
4020:
4004:
4001:
3996:
3992:
3986:
3982:
3952:
3948:
3941:
3937:
3932:
3927:
3924:
3917:
3916:
3915:
3913:
3894:
3891:
3888:
3880:
3873:
3872:
3871:
3869:
3855:
3853:
3849:
3845:
3826:
3820:
3817:
3812:
3808:
3804:
3801:
3798:
3793:
3790:
3787:
3783:
3773:
3768:
3765:
3761:
3753:
3752:
3751:
3749:
3745:
3741:
3737:
3733:
3729:
3719:
3717:
3698:
3692:
3686:
3683:
3677:
3670:
3667:
3659:
3658:
3657:
3655:
3651:
3646:
3642:
3618:
3614:
3610:
3605:
3602:
3599:
3595:
3584:
3581:
3578:
3574:
3568:
3564:
3559:
3555:
3547:
3543:
3539:
3532:
3528:
3523:
3519:
3516:
3513:
3508:
3504:
3500:
3493:
3489:
3484:
3480:
3475:
3471:
3467:
3460:
3456:
3451:
3447:
3442:
3439:
3436:
3432:
3428:
3421:
3418:
3415:
3411:
3406:
3392:
3391:
3390:
3385:
3378:
3371:
3364:
3357:
3350:
3343:
3339:
3335:
3325:
3321:
3317:
3313:
3309:
3305:
3301:
3296:
3292:
3285:
3278:
3271:
3267:
3262:
3258:
3248:
3246:
3242:
3224:
3221:
3217:
3208:
3204:
3186:
3183:
3179:
3158:
3152:
3146:
3143:
3140:
3135:
3132:
3128:
3124:
3119:
3116:
3112:
3108:
3102:
3099:
3093:
3087:
3084:
3081:
3078:
3072:
3069:
3066:
3060:
3046:
3042:
3038:
3021:
3017:
3014:
3011:
3008:
3003:
2999:
2994:
2973:
2970:
2965:
2962:
2959:
2955:
2934:
2931:
2926:
2922:
2913:
2909:
2905:
2887:
2883:
2869:
2860:
2857:
2854:
2850:
2846:
2842:
2837:
2833:
2829:
2826: â
2825:
2821:
2818:
2814:
2808:
2802:
2798:
2792:
2766:
2760:
2757:
2754:
2751:
2748:
2744:
2740:
2737:
2734:
2729:
2726:
2723:
2719:
2715:
2710:
2706:
2701:
2697:
2692:
2688:
2679:
2675:
2654:
2650:
2621:
2617:
2605:
2585:
2582:
2579:
2566:
2563:
2560:
2556:
2552:
2547:
2544:
2541:
2537:
2533:
2530:
2527:
2522:
2519:
2516:
2512:
2508:
2503:
2500:
2497:
2493:
2489:
2484:
2481:
2478:
2474:
2470:
2465:
2462:
2459:
2455:
2451:
2446:
2442:
2438:
2433:
2429:
2420:
2415:
2403:
2399:
2395:
2390:
2386:
2382:
2379:
2376:
2371:
2368:
2365:
2361:
2357:
2352:
2349:
2346:
2342:
2338:
2333:
2330:
2327:
2323:
2319:
2314:
2311:
2308:
2304:
2300:
2295:
2291:
2287:
2282:
2278:
2269:
2253:
2249:
2245:
2228:
2224:
2215:
2211:
2190:
2186:
2182:
2177:
2174:
2171:
2167:
2163:
2160:
2157:
2152:
2148:
2144:
2139:
2136:
2133:
2129:
2125:
2120:
2116:
2112:
2107:
2103:
2093:
2085:
2081:
2077:
2072:
2068:
2064:
2061:
2058:
2053:
2049:
2045:
2040:
2036:
2032:
2027:
2023:
2019:
2014:
2010:
1995:
1992:
1988:
1969:
1966:
1961:
1958:
1955:
1951:
1947:
1944:
1941:
1936:
1932:
1922:
1916:
1913:
1908:
1904:
1900:
1897:
1894:
1889:
1886:
1883:
1879:
1864:
1863:
1857:
1855:
1852:
1851:countable set
1847:
1843:
1824:
1821:
1813:
1809:
1805:
1800:
1796:
1792:
1789:
1786:
1781:
1777:
1773:
1768:
1764:
1749:
1733:
1725:
1721:
1717:
1712:
1708:
1704:
1701:
1698:
1693:
1690:
1687:
1683:
1673:
1665:
1661:
1657:
1652:
1648:
1644:
1641:
1638:
1633:
1629:
1625:
1620:
1616:
1612:
1607:
1603:
1599:
1594:
1590:
1586:
1583:
1580:
1575:
1572:
1569:
1565:
1551:
1550:
1549:
1547:
1540:
1533:
1526:
1523:
1517:
1502:
1488:
1485:
1482:
1479:
1476:
1473:
1468:
1465:
1462:
1458:
1437:
1434:
1431:
1428:
1425:
1422:
1417:
1413:
1391:
1388:
1385:
1382:
1379:
1376:
1371:
1367:
1344:
1340:
1317:
1313:
1292:
1289:
1286:
1283:
1280:
1277:
1272:
1268:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1208:
1205:
1202:
1199:
1196:
1193:
1188:
1184:
1175:
1157:
1153:
1144:
1126:
1122:
1112:
1096:
1092:
1069:
1065:
1042:
1038:
1017:
1011:
1006:
1002:
979:
975:
954:
948:
943:
939:
929:
927:
903:
900:
897:
892:
888:
864:
861:
856:
852:
825:
821:
801:
797:
793:
790:
786:
782:
777:
773:
770:
769:
767:
757:
755:
754:Eugene Dynkin
751:
747:
743:
739:
734:
730:
728:
724:
719:
715:
711:
707:
703:
702:finite groups
699:
694:
692:
688:
685:, written by
684:
683:Eugene Onegin
679:
675:
671:
666:
665:Andrey Markov
657:
653:
651:
647:
643:
637:
634:
624:
622:
618:
614:
609:
607:
603:
599:
588:
584:
580:
577:
573:
569:
565:
562:
559:
555:
545:
542:
539:
536:The system's
529:
526:
522:
517:
515:
511:
507:
503:
499:
487:
486:Andrey Markov
482:
473:
471:
467:
462:
460:
456:
452:
448:
444:
440:
436:
432:
428:
424:
420:
416:
411:
409:
408:Andrey Markov
405:
401:
397:
393:
389:
385:
381:
377:
374:describing a
373:
369:
365:
353:
348:
346:
341:
339:
334:
333:
331:
330:
325:
322:
320:
317:
316:
315:
314:
309:
306:
304:
301:
299:
296:
294:
291:
289:
286:
284:
281:
280:
279:
278:
273:
270:
268:
265:
263:
260:
258:
255:
254:
253:
252:
245:
242:
240:
237:
235:
232:
230:
227:
225:
222:
220:
217:
215:
212:
210:
207:
206:
205:
202:
200:
197:
193:
190:
188:
185:
183:
180:
178:
175:
173:
170:
168:
165:
164:
163:
160:
156:
153:
152:
151:
148:
144:
141:
139:
136:
134:
131:
129:
126:
124:
121:
120:
119:
116:
114:
111:
109:
106:
105:
104:
103:
98:
95:
93:
92:Indeterminism
90:
86:
83:
82:
81:
78:
74:
71:
70:
69:
66:
65:
64:
63:
59:
55:
54:
51:
48:
47:
44:
40:
39:
32:
19:
15916:Graph theory
15797:Econometrics
15759:Wiener space
15647:ItĂŽ integral
15548:Inequalities
15437:Self-similar
15407:GaussâMarkov
15397:Exchangeable
15377:CĂ dlĂ g paths
15313:Risk process
15265:LIBOR market
15134:Random graph
15129:Random field
14941:Superprocess
14879:LĂ©vy process
14874:Jump process
14849:Hunt process
14685:Markov chain
14684:
14565:
14537:
14522:
14515:
14500:
14483:
14459:
14445:
14437:
14425:
14406:
14405:S. P. Meyn.
14395:
14363:
14326:
14301:
14286:
14265:
14249:
14245:
14236:
14229:
14194:
14191:Solar Energy
14190:
14169:. Retrieved
14165:
14152:
14140:. Retrieved
14128:
14124:
14114:
14095:
14091:
14069:
14065:
14031:
14023:
14014:
14010:
14000:
13992:the original
13982:
13971:. Retrieved
13961:
13950:. Retrieved
13946:the original
13935:
13927:
13911:
13903:the original
13893:
13878:
13873:
13854:
13848:
13834:(2): 19â30.
13831:
13827:
13821:
13766:
13762:
13752:
13741:the original
13736:
13724:
13713:the original
13708:
13695:
13668:
13664:
13658:
13644:(1): 27â58.
13641:
13635:
13625:
13590:
13587:Econometrica
13586:
13580:
13571:
13567:
13560:
13533:
13529:
13523:
13514:
13510:
13504:
13479:
13475:
13469:
13450:
13444:
13409:
13405:
13392:
13372:
13365:
13352:
13332:
13318:
13304:
13261:
13257:
13247:
13231:
13227:
13221:
13196:
13192:
13182:
13157:
13154:Solar Energy
13153:
13147:
13122:
13119:Solar Energy
13118:
13112:
13087:
13084:Solar Energy
13083:
13077:
13050:
13047:Solar Energy
13046:
13036:
13011:
13008:Solar Energy
13007:
13001:
12976:
12973:Solar Energy
12972:
12966:
12923:
12919:
12909:
12864:
12860:
12850:
12817:
12813:
12806:
12781:
12777:
12771:
12746:
12742:
12736:
12683:
12679:
12669:
12643:
12637:
12609:
12602:
12567:
12563:
12539:. Retrieved
12526:
12515:
12474:
12465:
12456:
12443:
12424:
12418:
12399:
12395:
12385:
12365:
12358:
12347:. Retrieved
12345:. 2020-03-22
12342:
12333:
12313:
12306:
12295:. Retrieved
12291:
12256:
12250:
12238:. Retrieved
12222:
12211:. Retrieved
12207:
12200:Peres, Yuval
12194:
12175:
12169:
12158:. Retrieved
12143:
12133:
12114:
12108:
12098:
12065:
12061:
12055:
12044:. Retrieved
12024:
12020:
12010:
11991:
11985:
11974:. Retrieved
11962:
11935:
11899:
11890:
11870:
11863:
11853:
11846:
11826:
11819:
11799:
11792:
11765:
11759:
11739:
11709:
11682:
11678:
11671:
11651:
11644:
11619:
11615:
11609:
11589:
11582:
11562:
11555:
11536:
11532:
11502:
11498:
11459:
11449:
11424:
11420:
11414:
11389:
11385:
11376:
11359:
11355:
11349:
11312:
11306:
11286:
11262:(2): 92â96.
11259:
11255:
11224:
11188:
11164:
11159:
11144:
11139:
11123:
11118:
11102:
11097:
11082:
11077:
11057:
11050:
11030:
11023:
11003:
10996:
10976:
10969:
10949:
10912:
10906:
10883:
10872:
10852:
10845:
10825:
10818:
10798:
10673:
10649:
10628:
10619:
10616:
10607:
10603:
10601:
10335:
10319:
10298:middle class
10271:
10260:
10253:
10222:
10202:
10193:
9985:
9912:
9908:
9904:
9900:
9892:
9888:
9884:
9878:
9867:
9845:
9834:
9794:
9788:
9773:
9757:
9745:
9723:Neurobiology
9685:
9676:superlattice
9673:
9661:
9653:
9646:
9641:
9636:
9633:
9494:
9483:
9475:
9472:Applications
9429:
9425:
9422:finite graph
9415:
9396:
9390:
9378:
9321:
9318:Markov model
9312:Markov model
9300:
9296:
9292:
9288:
9284:
9282:
9269:
9265:
9263:
9058:
9000:
8995:
8983:
8975:
8973:
8898:
8896:
8759:
8755:
8746:
8742:
8738:
8734:jump process
8732:
8728:
8724:
8710:
8696:
8694:
8629:
8625:
8623:
8472:
8468:
8428:
8424:
8420:
8418:
8408:hitting time
8407:
8405:
8384:
8379:
8375:
8373:
8261:
8257:
8253:
8249:
8247:
8237:
8233:
8231:
8113:
8108:
8106:
7896:
7397:
7285:
7281:
7279:
7269:
7218:
7216:
7205:
7197:
7192:
7188:
7184:
7182:
7132:
7127:
7123:
7119:
7117:
7052:
7049:
7039:
7035:
7033:
7027:
6996:
6992:
6990:
6877:hitting time
6876:
6872:
6868:
6864:
6860:
6856:
6852:
6848:
6846:
6841:
6814:; otherwise
6785:
6783:
6680:
6674:
6664:
6658:
6652:
6649:
6644:
6640:
6635:
6633:
6623:
6609:
6601:
6594:
6571:
6564:
6557:
6481:
6474:
6470:
6464:
6457:
6451:
6445:
6439:
6432:
6426:
6424:
5602:
5598:
5594:
5590:
5583:
5577:
5571:
5565:
5561:
5557:
5555:
5412:
5408:
5404:
5403:be a length
5400:
5395:
5390:
5385:
5381:
5377:
5373:
5368:
5364:
5358:
5353:
5351:
5224:
5163:
5159:
5152:
5145:
5138:
5134:
5130:
5126:
5122:
5118:
5116:
5111:
5099:
5095:
5091:
5087:
5078:
5038:
5028:
5024:
5020:
5016:
5012:
5007:
5003:
4999:
4995:
4993:
4873:
4869:
4865:
4863:
4858:
4854:
4850:
4846:
4842:
4838:
4834:
4830:
4822:
4818:
4814:
4805:
4801:
4797:
4793:
4784:
4780:
4778:
4698:
4697:Subtracting
4696:
4654:
4594:
4590:
4586:
4583:
4580:
4284:, the limit
4281:
4279:
4225:
4223:
4155:
4149:
4144:
4140:
4136:
4132:
4130:
4063:
4032:
4022:
4018:
3973:
3909:
3867:
3861:
3847:
3843:
3841:
3747:
3739:
3735:
3725:
3713:
3644:
3640:
3638:
3383:
3376:
3369:
3362:
3355:
3348:
3341:
3337:
3333:
3331:
3319:
3315:
3311:
3307:
3303:
3298:follows the
3294:
3290:
3283:
3276:
3269:
3265:
3260:
3256:
3254:
3244:
3240:
3205:, using the
3044:
3043:and for all
3040:
3039:â 0 for all
3036:
2911:
2907:
2903:
2874:
2858:
2852:
2848:
2844:
2840:
2835:
2831:
2827:
2823:
2819:
2812:
2806:
2800:
2796:
2794:
2677:
2673:
2603:
2251:
2247:
2213:
2209:
1990:
1986:
1853:
1845:
1841:
1839:
1538:
1531:
1524:
1519:
1173:
1142:
1113:
930:
925:
844:draws, with
811:
772:Random walks
731:
695:
663:
654:
638:
630:
610:
606:Markov model
601:
597:
594:
568:Harris chain
535:
518:
495:
469:
465:
463:
412:
383:
367:
364:Markov chain
363:
361:
324:Tree diagram
319:Venn diagram
283:Independence
229:Markov chain
228:
113:Sample space
15842:Ruin theory
15780:Disciplines
15652:ItĂŽ's lemma
15427:Predictable
15102:Percolation
15085:Potts model
15080:Ising model
15044:White noise
15002:Differences
14864:ItĂŽ process
14804:Cox process
14700:Loop-erased
14695:Random walk
14482:Seneta, E.
14266:Probability
14197:: 688â695.
14131:(1): 7â14.
14098:: 152â158.
13160:: 174â183.
13090:: 487â495.
13053:: 229â242.
13014:: 160â170.
12518:, Springer
12021:SIAM Review
11113:(Table 6.1)
10879:"Markovian"
10284:Das Kapital
9881:M/M/1 queue
9763:solar power
9497:lattice QCD
9278:unit vector
8897:From this,
8758:into state
8624:For a CTMC
8390:time series
8234:irreducible
8109:irreducible
7202:Terminology
6875:, the mean
6684:. That is:
6656:has period
6645:irreducible
6636:communicate
5399:. Also let
5167:). Then by
5133:, that is,
4811:zero matrix
4117:dot product
3912:eigenvector
2676:-tuples of
1143:total value
796:number line
636:terminate.
627:Transitions
613:state space
538:state space
521:state space
514:independent
510:conditional
380:probability
239:Random walk
80:Determinism
68:Probability
15900:Categories
15852:Statistics
15632:Filtration
15533:Kolmogorov
15517:Blumenthal
15442:Stationary
15382:Continuous
15370:Properties
15255:HullâWhite
14997:Martingale
14884:Local time
14772:Fractional
14750:pure birth
14335:. Berlin:
14312:. online:
14283:J. L. Doob
14222:References
13973:2009-04-24
13952:2007-11-26
13887:1576470792
12541:2017-06-02
12459:: 781â784.
12349:2024-02-01
12297:2024-02-01
12292:bactra.org
12213:2024-02-01
12160:2016-03-04
12046:2021-05-31
11976:2017-06-02
11382:Seneta, E.
11143:Dodge, Y.
10922:3540047581
10900:required.)
10294:capitalism
10199:Statistics
9703:nucleotide
9664:copolymers
9299:(ÎŽ),
9295:(0),
9021:such that
8697:reversible
8138:such that
7490:goes like
7114:Ergodicity
7038:is called
6869:persistent
6630:Properties
4825:must be a
4027:eigenvalue
1860:Variations
621:Variations
492:Definition
476:Principles
394:(DTMC). A
150:Experiment
97:Randomness
43:statistics
15764:Classical
14777:Geometric
14767:Excursion
14572:EMS Press
14400:Fizmatgiz
14142:5 October
13779:CiteSeerX
13673:CiteSeerX
13671:: 49â83.
13595:CiteSeerX
13536:: 21â86.
13517:: 425â40.
13455:CiteSeerX
13422:CiteSeerX
13234:: 81â91,
13213:1134-3060
12693:1209.6210
11996:CiteSeerX
11685:(1): 57.
11505:(1): 33.
11317:CiteSeerX
10640:AstroTurf
10302:political
10278:Karl Marx
10258:setting.
10247:model of
10179:α
10153:α
10150:−
10118:α
10115:−
10091:α
9964:α
9961:−
9937:α
9781:systems.
9656:in silico
9599:Catalytic
9591:⟶
9576:Substrate
9562:⇀
9555:−
9544:−
9537:↽
9503:Chemistry
9228:−
9211:
9202:φ
9187:−
9170:
9161:φ
9158:−
9149:π
9115:‖
9111:φ
9108:‖
9087:φ
9067:φ
9041:φ
9032:φ
9009:φ
8986:) is the
8982:and diag(
8951:−
8933:
8921:−
8871:otherwise
8855:≠
8821:≠
8814:∑
8671:−
8648:^
8594:∉
8542:∈
8535:∑
8531:−
8521:∈
8315:∈
8217:Ω
8197:∅
8151:−
8104:instead.
8084:Σ
8077:Ω
8054:…
8029:…
7977:Ω
7974:→
7971:Ω
7944:Σ
7916:Σ
7909:Ω
7846:−
7810:≤
7760:−
7742:≤
7716:−
7707:≤
7614:−
7608:−
7599:≤
7576:≥
7519:→
7513:→
7510:⋯
7507:→
7501:→
7439:−
7430:≤
7219:primitive
7062:π
7040:absorbing
6949:⋅
6941:∞
6926:∑
6865:recurrent
6853:transient
6842:aperiodic
6741:∣
6530:λ
6521:≥
6518:⋯
6515:≥
6501:λ
6364:λ
6354:λ
6330:⋯
6287:λ
6277:λ
6216:λ
6206:λ
6137:λ
6120:≠
6099:⊥
6053:λ
6036:⋯
6000:λ
5953:λ
5924:−
5907:Σ
5862:⋯
5775:−
5758:Σ
5728:−
5720:Σ
5708:⋯
5696:−
5688:Σ
5667:−
5659:Σ
5622:π
5533:∈
5477:∑
5323:λ
5314:≥
5311:⋯
5308:≥
5294:λ
5285:≥
5271:λ
5248:λ
5202:−
5194:Σ
5056:π
5048:π
4959:−
4936:−
4725:−
4624:∞
4621:→
4303:∞
4300:→
4251:∞
4248:→
4209:π
4184:∞
4181:→
4091:π
4087:⋅
4075:∑
4043:π
3993:π
3983:∑
3938:∑
3925:π
3892:π
3881:π
3805:∣
3750:equal to
3611:−
3517:…
3448:∣
3247:happens.
3180:δ
3113:δ
3085:∣
3035:, and as
2843:to state
2752:−
2738:…
2727:−
2564:−
2545:−
2531:…
2520:−
2501:−
2482:−
2463:−
2452:∣
2380:…
2369:−
2350:−
2331:−
2312:−
2301:∣
2161:…
2062:…
1959:−
1948:∣
1901:∣
1790:…
1705:∣
1642:…
1587:∣
1477:ℓ
1466:−
1238:×
1232:×
1015:$
1012:≥
952:$
904:∈
466:Markovian
439:economics
435:chemistry
143:Singleton
15885:Category
15769:Abstract
15303:BĂŒhlmann
14909:Compound
14582:Archived
14323:(1965).
14171:26 March
13920:Archived
13813:22198760
13341:Archived
13296:25984837
12958:27429455
12901:19816557
12842:22186291
12798:19527020
12763:19527020
12728:23408514
12594:26968853
12230:(2016).
12090:22181092
11456:Heyde CC
10931:52203046
10686:See also
10625:Baseball
10438:♭
10419:♯
10395:♭
10387:♯
10344:such as
10342:software
10288:, tying
9988:PageRank
9812:through
9237:‖
9198:‖
8848:if
8719:, of an
8183:implies
7936:, where
7864:, where
7286:exponent
7118:A state
7034:A state
6847:A state
6786:periodic
6650:A state
5125:and let
4813:of size
4792:of size
4025:with an
3671:′
2910:at time
2250:) where
2208:for all
1985:for all
1746:if both
760:Examples
696:In 1912
642:integers
376:sequence
224:Variance
15392:Ergodic
15280:VaĆĄĂÄek
15122:Poisson
14782:Meander
14597:YouTube
14574:, 2001
14458:(1960)
14285:(1953)
14199:Bibcode
13804:3271566
13771:Bibcode
13617:1912559
13496:2227127
13414:Bibcode
13287:4434998
13266:Bibcode
13162:Bibcode
13127:Bibcode
13092:Bibcode
13055:Bibcode
13016:Bibcode
12981:Bibcode
12949:4946376
12928:Bibcode
12892:2749218
12869:Bibcode
12822:Bibcode
12719:3568780
12698:Bibcode
12585:5862921
12240:22 June
12070:Bibcode
12041:2132659
11624:Bibcode
11441:1403518
11406:1403785
11147:, OUP.
11126:, OUP.
11085:. CUP.
10632:bunting
10612:phrasal
9805:entropy
9742:Testing
9682:Biology
9579:binding
9480:Physics
9434:measure
8978:is the
8721:ergodic
8238:ergodic
7884:is the
7798:is the
7270:regular
7124:ergodic
6668:is the
5372:be the
5137:= diag(
4809:is the
4788:is the
4121:simplex
3744:element
3201:is the
1849:form a
1501:state.
660:History
451:physics
443:finance
431:biology
404:Russian
138:Outcome
15732:Tanaka
15417:Mixing
15412:Markov
15285:Wilkie
15250:HoâLee
15245:Heston
15017:Super-
14762:Bridge
14710:Biased
14544:
14529:
14507:
14490:
14476:
14466:
14454:&
14413:
14380:
14351:
14308:
14293:
14275:
14040:
13885:
13861:
13811:
13801:
13781:
13675:
13615:
13597:
13494:
13457:
13424:
13380:
13294:
13284:
13211:
12956:
12946:
12899:
12889:
12840:
12796:
12761:
12726:
12716:
12658:
12625:
12592:
12582:
12489:
12431:
12373:
12321:
12263:
12182:
12151:
12088:
12039:
11998:
11950:
11914:
11878:
11834:
11807:
11780:
11747:
11717:
11659:
11597:
11570:
11474:
11439:
11404:
11337:
11319:
11294:
11232:
11200:
11196:â466.
11151:
11130:
11109:
11089:
11065:
11038:
11011:
10984:
10957:
10929:
10919:
10860:
10833:
10806:
10678:, and
10352:, and
10346:Csound
9992:Google
9707:genome
9452:, the
9448:, the
8974:where
8107:Since
7778:where
6991:State
6641:closed
6556:hence
6425:Since
5352:Since
4796:, and
4779:where
4589:be an
4224:where
3732:matrix
3728:finite
3639:where
3209:. The
3171:where
470:Markov
457:, and
85:System
73:Axioms
15585:Tools
15361:M/M/c
15356:M/M/1
15351:M/G/1
15341:Fluid
15007:Local
14398:) by
14162:(PDF)
13744:(PDF)
13733:(PDF)
13716:(PDF)
13705:(PDF)
13613:JSTOR
13492:JSTOR
13402:(PDF)
12688:arXiv
12536:(PDF)
12453:(PDF)
12235:(PDF)
12037:JSTOR
11971:(PDF)
11437:JSTOR
11402:JSTOR
10894:
10780:Notes
10582:0.25
10540:0.75
10484:0.22
10461:Notes
10332:Music
10324:and "
10316:Games
9428:or a
9420:of a
9059:with
8688:. By
7284:, or
5560:with
5416:span
5158:,...,
4990:ones.
3850:is a
2639:from
1174:count
386:." A
370:is a
118:Event
15537:LĂ©vy
15336:Bulk
15220:Chen
15012:Sub-
14970:Both
14542:ISBN
14527:ISBN
14505:ISBN
14488:ISBN
14474:ISBN
14464:ISBN
14419:CTCN
14411:ISBN
14378:ISBN
14349:ISBN
14314:MCSS
14306:ISBN
14291:ISBN
14273:ISBN
14173:2024
14144:2023
14038:ISBN
14011:BYTE
13883:ISBN
13859:ISBN
13809:PMID
13564:e.g.
13378:ISBN
13292:PMID
13209:ISSN
12954:PMID
12897:PMID
12838:PMID
12794:PMID
12759:PMID
12724:PMID
12656:ISBN
12623:ISBN
12590:PMID
12487:ISBN
12429:ISBN
12371:ISBN
12319:ISBN
12261:ISBN
12242:2024
12180:ISBN
12149:ISBN
12086:PMID
11948:ISBN
11912:ISBN
11876:ISBN
11832:ISBN
11805:ISBN
11778:ISBN
11745:ISBN
11715:ISBN
11657:ISBN
11595:ISBN
11568:ISBN
11472:ISBN
11335:ISBN
11292:ISBN
11230:ISBN
11198:ISBN
11149:ISBN
11128:ISBN
11107:ISBN
11087:ISBN
11063:ISBN
11036:ISBN
11009:ISBN
10982:ISBN
10955:ISBN
10927:OCLC
10917:ISBN
10858:ISBN
10831:ISBN
10804:ISBN
10634:and
10579:0.25
10568:0.2
10534:0.25
10512:0.1
10509:0.75
10506:0.15
10478:0.18
10430:0.7
10427:0.05
10424:0.25
10411:0.3
10378:Note
10363:MIDI
10054:has
9986:The
9848:LZMA
9846:The
9824:and
9693:and
9602:step
9488:and
9464:and
9208:diag
9167:diag
8930:diag
8406:The
7565:has
7280:The
6867:(or
6799:>
6763:>
6710:>
6626:or.
6618:and
5262:>
5117:Let
5098:has
3742:)th
3015:<
2875:Let
2815:, a
2583:>
2212:and
1822:>
1018:0.60
955:0.50
803:one.
716:and
708:and
706:Paul
640:the
581:Any
468:and
15117:Cox
14595:on
14370:doi
14341:doi
14254:doi
14207:doi
14195:184
14133:doi
14100:doi
14096:122
13836:doi
13799:PMC
13789:doi
13767:108
13683:doi
13646:doi
13642:105
13605:doi
13546:hdl
13538:doi
13484:doi
13432:doi
13282:PMC
13274:doi
13236:doi
13201:doi
13170:doi
13158:170
13135:doi
13100:doi
13088:173
13063:doi
13051:115
13024:doi
13012:103
12989:doi
12944:PMC
12936:doi
12887:PMC
12877:doi
12830:doi
12786:doi
12751:doi
12714:PMC
12706:doi
12648:doi
12615:doi
12580:PMC
12572:doi
12514:",
12479:doi
12404:doi
12400:158
12119:doi
12078:doi
12029:doi
11940:doi
11904:doi
11770:doi
11687:doi
11632:doi
11541:doi
11507:doi
11464:doi
11429:doi
11394:doi
11364:doi
11327:doi
11264:doi
11260:101
11194:464
10604:and
10587:GD
10576:0.5
10573:GA
10565:0.4
10562:0.4
10559:GG
10551:0.1
10548:0.9
10545:DG
10531:DA
10517:DD
10503:AG
10495:0.5
10492:0.5
10489:AD
10481:0.6
10475:AA
10446:0.3
10443:0.7
10408:0.6
10405:0.1
10350:Max
10308:to
10280:'s
9903:to
9887:to
9872:).
9444:of
9397:any
9280:.)
8374:If
8209:or
8114:In
7632:If
7545:If
6999:if
6995:is
6788:if
6701:gcd
6662:if
6469:as
6463:=
5601:as
5593:...
5591:xPP
5110:of
4843:n+1
4614:lim
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2509:=
2504:2
2498:n
2494:X
2490:,
2485:1
2479:n
2475:x
2471:=
2466:1
2460:n
2456:X
2447:n
2443:x
2439:=
2434:n
2430:X
2426:(
2416:=
2409:)
2404:1
2400:x
2396:=
2391:1
2387:X
2383:,
2377:,
2372:2
2366:n
2362:x
2358:=
2353:2
2347:n
2343:X
2339:,
2334:1
2328:n
2324:x
2320:=
2315:1
2309:n
2305:X
2296:n
2292:x
2288:=
2283:n
2279:X
2275:(
2252:m
2248:m
2229:0
2225:X
2214:k
2210:n
2196:)
2191:k
2187:x
2183:=
2178:k
2175:+
2172:n
2168:X
2164:,
2158:,
2153:1
2149:x
2145:=
2140:1
2137:+
2134:n
2130:X
2126:,
2121:0
2117:x
2113:=
2108:n
2104:X
2100:(
2094:=
2091:)
2086:k
2082:x
2078:=
2073:k
2069:X
2065:,
2059:,
2054:1
2050:x
2046:=
2041:1
2037:X
2033:,
2028:0
2024:x
2020:=
2015:0
2011:X
2007:(
1993:.
1991:n
1987:n
1973:)
1970:y
1967:=
1962:1
1956:n
1952:X
1945:x
1942:=
1937:n
1933:X
1929:(
1923:=
1920:)
1917:y
1914:=
1909:n
1905:X
1898:x
1895:=
1890:1
1887:+
1884:n
1880:X
1876:(
1854:S
1846:i
1842:X
1819:)
1814:n
1810:x
1806:=
1801:n
1797:X
1793:,
1787:,
1782:1
1778:x
1774:=
1769:1
1765:X
1761:(
1734:,
1731:)
1726:n
1722:x
1718:=
1713:n
1709:X
1702:x
1699:=
1694:1
1691:+
1688:n
1684:X
1680:(
1674:=
1671:)
1666:n
1662:x
1658:=
1653:n
1649:X
1645:,
1639:,
1634:2
1630:x
1626:=
1621:2
1617:X
1613:,
1608:1
1604:x
1600:=
1595:1
1591:X
1584:x
1581:=
1576:1
1573:+
1570:n
1566:X
1562:(
1542:3
1539:X
1535:2
1532:X
1528:1
1525:X
1489:p
1486:,
1483:m
1480:,
1474:=
1469:1
1463:n
1459:X
1438:k
1435:,
1432:j
1429:,
1426:i
1423:=
1418:n
1414:X
1392:1
1389:,
1386:0
1383:,
1380:1
1377:=
1372:2
1368:X
1345:1
1341:X
1318:2
1314:X
1293:0
1290:,
1287:1
1284:,
1281:0
1278:=
1273:1
1269:X
1244:=
1241:6
1235:6
1229:6
1209:5
1206:,
1203:0
1200:,
1197:1
1194:=
1189:6
1185:X
1158:n
1154:X
1127:n
1123:X
1097:6
1093:X
1070:7
1066:X
1043:6
1039:X
1007:7
1003:X
980:6
976:X
949:=
944:6
940:X
912:}
908:N
901:n
898::
893:n
889:X
885:{
865:0
862:=
857:0
853:X
842:n
826:n
822:X
351:e
344:t
337:v
20:)
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