27:
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1185:
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3130:– Each outcome of the card will not affect the next one, which means the 52 results are independent from each other. In contrast, if each card that is drawn is kept out of the deck, subsequent draws would be affected by it (drawing one king would make drawing a second king less likely), and the result would not be independent.
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The second reason is that the accuracy of the model depends on the simplicity and representational power of the model unit, as well as the quality of the data. The simplicity of the unit makes it easy to interpret and scale, while the representational power and scalability improve model accuracy. In
3832:
Even if the sample originates from a complex non-Gaussian distribution, it can be well-approximated because the central limit theorem allows it to be simplified to a
Gaussian distribution. For a large number of observable samples, "the sum of many random variables will have an approximately normal
1073:
3245:
utilizes the vast amounts of data currently available to deliver faster and more accurate results. To train machine learning models effectively, it is crucial to use historical data that is broadly generalizable. If the training data is not representative of the overall situation, the model's
1911:
3256:
This assumption simplifies mathematical maximization calculations. In optimization problems, the assumption of independent and identical distribution simplifies the calculation of the likelihood function. Due to the independence assumption, the likelihood function can be expressed as:
3824:
Computers are very efficient at performing multiple additions, but not as efficient at performing multiplications. This simplification enhances computational efficiency. The log transformation, in the process of maximizing, converts many exponential functions into linear functions.
2763:{\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\definecolor {blue}{rgb}{0,0,1}\definecolor {Blue}{rgb}{0,0,1}P({\color {red}A}{\color {green}B}{\color {blue}C})=P({\color {red}A})P({\color {green}B})P({\color {blue}C})}
1501:{\displaystyle {\begin{aligned}&F_{X_{1}}(x)=F_{X_{k}}(x)\,&\forall k\in \{1,\ldots ,n\}{\text{ and }}\forall x\in I\\&F_{X_{1},\ldots ,X_{n}}(x_{1},\ldots ,x_{n})=F_{X_{1}}(x_{1})\cdot \ldots \cdot F_{X_{n}}(x_{n})\,&\forall x_{1},\ldots ,x_{n}\in I\end{aligned}}}
2332:
174:
assumption frequently arises in the context of sequences of random variables. Then, "independent and identically distributed" implies that an element in the sequence is independent of the random variables that came before it. In this way, an i.i.d. sequence is different from a
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2015:
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2590:{\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\definecolor {blue}{rgb}{0,0,1}\definecolor {Blue}{rgb}{0,0,1}P({\color {green}B}{\color {blue}C})=P({\color {green}B})P({\color {blue}C})}
3114:– Regardless of whether the die is fair or weighted, each roll will have the same probability as every other roll. In contrast, rolling 10 different dice, some of which are weighted and some of which are not, would not produce i.i.d. variables.
2192:
1916:
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means that the sample items are all independent events. In other words, they are not connected to each other in any way; knowledge of the value of one variable gives no information about the value of the other and vice
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1866:{\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}P({\color {red}A}\ \mathrm {and} \ {\color {green}B})=P({\color {red}A})P({\color {green}B})}
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th random variable is a function of the previous random variable in the sequence (for a first-order Markov sequence). An i.i.d. sequence does not imply the probabilities for all elements of the
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671:
3060:. One implication of this is that if the roulette ball lands on "red", for example, 20 times in a row, the next spin is no more or less likely to be "black" than on any other spin (see the
3085:– Regardless of whether the coin is fair (probability 1/2 of heads) or unfair, as long as the same coin is used for each flip, each flip will have the same probability as each other flip.
2600:
2515:{\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}P({\color {red}A}{\color {green}B})=P({\color {red}A})P({\color {green}B})}
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are mutually independent which cannot be established with mutually incompatible at the same time; that is, independence must be compatible and mutual exclusion must be related.
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Independent and identically distributed random variables are often used as an assumption, which tends to simplify the underlying mathematics. In practical applications of
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2246:{\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}P({\color {green}B}|{\color {red}A})=P({\color {green}B})}
3175:. Exchangeability means that while variables may not be independent, future ones behave like past ones — formally, any value of a finite sequence is as likely as any
3837:
a deep neural network, for instance, each neuron is simple yet powerful in representation, layer by layer, capturing more complex features to enhance model accuracy.
3123:
Choose a card from a standard deck of cards containing 52 cards, then place the card back in the deck. Repeat this 52 times. Record the number of kings that appear.
1963:{\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}P({\color {red}A}\ \mathrm {and} \ {\color {green}B})}
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92:
Statistics commonly deals with random samples. A random sample can be thought of as a set of objects that are chosen randomly. More formally, it is "a sequence of
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3253:, or independent and identically distributed hypothesis, allows for a significant reduction in the number of individual cases required in the training sample.
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1068:{\displaystyle {\begin{aligned}&F_{X}(x)=F_{Y}(x)\,&\forall x\in I\\&F_{X,Y}(x,y)=F_{X}(x)\cdot F_{Y}(y)\,&\forall x,y\in I\end{aligned}}}
189:
or event space must be the same. For example, repeated throws of loaded dice will produce a sequence that is i.i.d., despite the outcomes being biased.
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1906:{\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}P({\color {red}A}{\color {green}B})}
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2327:{\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}P({\color {Green}B})>0}
5415:
3219:: each variable gives how much one changes from one time to another. For example, a sequence of Bernoulli trials is interpreted as the
4982:
3483:{\displaystyle l(\theta )=P(x_{1},x_{2},x_{3},...,x_{n}|\theta )=P(x_{1}|\theta )P(x_{2}|\theta )P(x_{3}|\theta )...P(x_{n}|\theta )}
4666:
3136:– After drawing one card from it, each time the probability for a king is 4/52, which means the probability is identical each time.
1676:{\displaystyle F_{X_{1},\ldots ,X_{n}}(x_{1},\ldots ,x_{n})=\operatorname {P} (X_{1}\leq x_{1}\land \ldots \land X_{n}\leq x_{n})}
4992:
4676:
124:
means that there are no overall trends — the distribution does not fluctuate and all items in the sample are taken from the same
3813:{\displaystyle \log(l(\theta ))=\log(P(x_{1}|\theta ))+\log(P(x_{2}|\theta ))+\log(P(x_{3}|\theta ))+...+\log(P(x_{n}|\theta ))}
5034:
4931:
2817:
2428:
2807:{\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\color {Green}B}
2418:{\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\color {Green}B}
2371:{\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\color {Green}B}
2182:{\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\color {Green}B}
2138:{\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\color {Green}B}
2010:{\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\color {Green}B}
1774:{\textstyle \definecolor {Green}{rgb}{0,0.5019607843137255,0}\definecolor {green}{rgb}{0,0.5019607843137255,0}\color {Green}B}
5221:
5211:
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5126:
3108:– Each outcome of the die roll will not affect the next one, which means the 10 results are independent from each other.
3079:– Each outcome of landing will not affect the other outcome, which means the 10 results are independent from each other.
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4236:
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55:
2665:{\textstyle \definecolor {blue}{rgb}{0,0,1}P({\color {red}A}{\color {blue}C})=P({\color {red}A})P({\color {blue}C})}
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To maximize the probability of the observed event, the log function is applied to maximize the parameter
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2020:
1686:
1120:
159:, which states that the probability distribution of the sum (or average) of i.i.d. variables with finite
5111:
200:, the notion of transformation to i.i.d. implies two specifications, the "i.d." part and the "i." part:
5116:
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There are two main reasons why this hypothesis is practically useful with the central limit theorem:
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76:. IID was first defined in statistics and finds application in different fields such as
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Many results that were first proven under the assumption that the random variables are
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3230:, and many Lévy processes can be seen as limits of i.i.d. variables—for instance, the
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events are equal to the product of the probabilities of each event, then the events
2145:— this is called conditional probability. Additionally, only when the occurrence of
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212:. – The signal spectrum must be flattened, i.e. transformed by filtering (such as
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3918:"IID Statistics: Independent and Identically Distributed Definition and Examples"
3184:
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The most general notion which shares the main properties of i.i.d. variables are
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3032:{\textstyle {\color {red}A}_{1},{\color {red}A}_{2},\ldots ,{\color {red}A}_{n}}
2923:{\textstyle {\color {red}A}_{1},{\color {red}A}_{2},\ldots ,{\color {red}A}_{n}}
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The definition extends naturally to more than two random variables. We say that
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113:" is the typical terminology, but in probability, it is more common to say "
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3583:{\displaystyle \mathop {\rm {argmax}} \limits _{\theta }\log(l(\theta ))}
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Toss a coin 10 times and record how many times the coin lands on heads.
5361:
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4845:
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4682:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
4122:
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1169:
Independence (probability theory) § More than two random variables
39:
4855:
824:{\displaystyle F_{X,Y}(x,y)=F_{X}(x)\cdot F_{Y}(y)\,\forall x,y\in I}
546:{\displaystyle F_{X,Y}(x,y)=\operatorname {P} (X\leq x\land Y\leq y)}
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160:
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signal (i.e. a signal where all frequencies are equally present).
3089:
Such a sequence of two possible i.i.d. outcomes is also called a
26:
1092:
3994:
Blum, J. R.; Chernoff, H.; Rosenblatt, M.; Teicher, H. (1958).
3101:
Roll a die 10 times and record how many times the result is 1.
3993:
833:
Independence (probability theory) § Two random variables
4662:
Autoregressive conditional heteroskedasticity (ARCH) model
2829:{\textstyle \definecolor {blue}{rgb}{0,0,1}\color {blue}C}
2440:{\textstyle \definecolor {blue}{rgb}{0,0,1}\color {blue}C}
4190:
Independent and identically distributed random variables
206:. – The signal level must be balanced on the time axis.
148:, however, this assumption may or may not be realistic.
4667:
Autoregressive integrated moving average (ARIMA) model
3996:"Central Limit Theorems for Interchangeable Processes"
3941:
Hampel, Frank (1998), "Is statistics too difficult?",
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denotes the joint cumulative distribution function of
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This provides a useful generalization — for example,
2930:. If the probabilities of the product events for any
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A sequence of outcomes of spins of a fair or unfair
414:{\displaystyle F_{Y}(y)=\operatorname {P} (Y\leq y)}
354:{\displaystyle F_{X}(x)=\operatorname {P} (X\leq x)}
3246:performance on new, unseen data may be inaccurate.
3226:One may generalize this to include continuous time
19:"IID" and "iid" redirect here. For other uses, see
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3149:. have been shown to be true even under a weaker
2094:{\textstyle P({\color {green}B}|{\color {red}A})}
666:{\displaystyle F_{X}(x)=F_{Y}(x)\,\forall x\in I}
5407:
4549:Stochastic chains with memory of variable length
3156:
1973:Suppose there are two events of the experiment,
1732:
16:Important notion in probability and statistics
4138:
4105:
4063:
3882:"A brief primer on probability distributions"
1175:identically distributed, i.e. if and only if
1093:Definition for more than two random variables
886:identically distributed, i.e. if and only if
179:, where the probability distribution for the
1279:
1261:
4034:
4677:Autoregressive–moving-average (ARMA) model
4145:
4131:
94:independent, identically distributed (IID)
58:. This property is usually abbreviated as
4011:
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3194:is not independent, but is exchangeable.
1453:
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4152:
4077:"What is Machine Learning? A Definition"
1781:, are called independent if and only if
25:
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3234:is the limit of the Bernoulli process.
3212:, i.i.d. variables are thought of as a
2839:A more general definition is there are
294:{\displaystyle I\subseteq \mathbb {R} }
48:independent and identically distributed
5408:
4983:Doob's martingale convergence theorems
3940:
3237:
467:joint cumulative distribution function
30:A chart showing a uniform distribution
4735:Constant elasticity of variance (CEV)
4725:Chan–Karolyi–Longstaff–Sanders (CKLS)
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1167:if they are independent (see further
50:if each random variable has the same
4035:Cover, T. M.; Thomas, J. A. (2006).
2288:{\textstyle P({\color {red}A})>0}
2123:has an effect on the probability of
2049:{\textstyle P({\color {red}A})>0}
1179:
889:
2167:has no effect on the occurrence of
2070:
1737:In probability theory, two events,
1722:{\displaystyle X_{1},\ldots ,X_{n}}
1156:{\displaystyle X_{1},\ldots ,X_{n}}
229:Definition for two random variables
54:as the others and all are mutually
13:
5222:Skorokhod's representation theorem
5003:Law of large numbers (weak/strong)
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233:Suppose that the random variables
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5192:Martingale representation theorem
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465:, respectively, and denote their
423:cumulative distribution functions
5237:Stochastic differential equation
5127:Doob's optional stopping theorem
5122:Doob–Meyer decomposition theorem
273:are defined to assume values in
109:are synonymous. In statistics, "
5107:Convergence of random variables
4993:Fisher–Tippett–Gnedenko theorem
4000:Canadian Journal of Mathematics
3197:
3039:are independent of each other.
2770:are satisfied, then the events
2101:. Generally, the occurrence of
155:assumption is also used in the
87:
4705:Binomial options pricing model
4069:
4057:
4037:Elements Of Information Theory
4028:
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3943:Canadian Journal of Statistics
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396:
384:
378:
348:
336:
324:
318:
139:
1:
5172:Kolmogorov continuity theorem
5008:Law of the iterated logarithm
3863:
3514:. Specifically, it computes:
3169:exchangeable random variables
3163:Exchangeable random variables
3157:Exchangeable random variables
223:
5177:Kolmogorov extension theorem
4856:Generalized queueing network
4364:Interacting particle systems
3192:sampling without replacement
3118:
3096:
3067:
3047:
7:
4309:Continuous-time random walk
3841:
3042:
1733:Definition for independence
10:
5437:
5317:Extreme value theory (EVT)
5117:Doob decomposition theorem
4409:Ornstein–Uhlenbeck process
4180:Chinese restaurant process
3201:
3160:
2961:{\textstyle 2,3,\ldots ,n}
2836:are mutually independent.
2785:{\textstyle \color {red}A}
2396:{\textstyle \color {red}A}
2349:{\textstyle \color {red}A}
2160:{\textstyle \color {red}A}
2116:{\textstyle \color {red}A}
1988:{\textstyle \color {red}A}
1752:{\textstyle \color {red}A}
99:In other words, the terms
18:
5385:
5289:
5197:Optional stopping theorem
5094:
5056:
4998:Large deviation principle
4965:
4879:
4836:
4803:
4750:Heath–Jarrow–Morton (HJM)
4695:
4687:Moving-average (MA) model
4672:Autoregressive (AR) model
4652:
4562:
4497:Hidden Markov model (HMM)
4479:
4431:Schramm–Loewner evolution
4235:
4160:
4117:, Duxbury Advanced Series
4064:Casella & Berger 2002
2056:, there is a possibility
5112:Doléans-Dade exponential
4942:Progressively measurable
4740:Cox–Ingersoll–Ross (CIR)
3916:Stephanie (2016-05-11).
882:if they are independent
52:probability distribution
5332:Mathematical statistics
5322:Large deviations theory
5152:Infinitesimal generator
5013:Maximal ergodic theorem
4932:Piecewise-deterministic
4534:Random dynamical system
4399:Markov additive process
3183:is invariant under the
3134:Identically distributed
3112:Identically distributed
3083:Identically distributed
598:identically distributed
122:Identically distributed
5167:Karhunen–Loève theorem
5102:Cameron–Martin formula
5066:Burkholder–Davis–Gundy
4461:Variance gamma process
4013:10.4153/CJM-1958-026-0
3814:
3584:
3508:
3484:
3179:of those values — the
3033:
2962:
2924:
2853:
2830:
2808:
2786:
2764:
2666:
2591:
2516:
2441:
2419:
2397:
2372:
2350:
2328:
2289:
2247:
2183:
2161:
2139:
2117:
2095:
2050:
2011:
1989:
1964:
1907:
1867:
1775:
1753:
1723:
1677:
1502:
1157:
1111:
1069:
872:
852:
825:
710:
690:
667:
590:
570:
547:
459:
439:
415:
355:
295:
267:
247:
31:
5297:Actuarial mathematics
5259:Uniform integrability
5254:Stratonovich integral
5182:Lévy–Prokhorov metric
5086:Marcinkiewicz–Zygmund
4973:Central limit theorem
4575:Gaussian random field
4404:McKean–Vlasov process
4324:Dyson Brownian motion
4185:Galton–Watson process
4115:Statistical Inference
4066:, Theorem 1.5.10
3858:Central limit theorem
3815:
3585:
3509:
3485:
3034:
2963:
2925:
2854:
2831:
2809:
2787:
2765:
2667:
2592:
2517:
2447:are three events. If
2442:
2420:
2398:
2373:
2351:
2329:
2290:
2248:
2184:
2162:
2140:
2118:
2096:
2051:
2012:
1990:
1965:
1908:
1868:
1776:
1754:
1724:
1678:
1503:
1158:
1112:
1070:
873:
853:
838:Two random variables
826:
711:
691:
676:Two random variables
668:
591:
571:
556:Two random variables
548:
460:
440:
416:
356:
296:
268:
248:
157:central limit theorem
96:random data points."
29:
5372:Time series analysis
5327:Mathematical finance
5212:Reflection principle
4539:Regenerative process
4339:Fleming–Viot process
4154:Stochastic processes
3848:De Finetti's theorem
3603:
3521:
3507:{\textstyle \theta }
3498:
3264:
2972:
2934:
2863:
2843:
2818:
2796:
2774:
2676:
2601:
2526:
2451:
2429:
2407:
2385:
2360:
2338:
2299:
2260:
2193:
2171:
2149:
2127:
2105:
2060:
2021:
1999:
1977:
1917:
1877:
1873:. In the following,
1785:
1763:
1741:
1687:
1529:
1186:
1121:
1101:
896:
862:
842:
724:
700:
680:
606:
580:
560:
473:
449:
429:
365:
305:
277:
257:
237:
146:statistical modeling
21:IID (disambiguation)
5367:Stochastic analysis
5207:Quadratic variation
5202:Prokhorov's theorem
5137:Feynman–Kac formula
4607:Markov random field
4255:Birth–death process
3964:20.500.11850/145503
3238:In machine learning
3210:stochastic calculus
165:normal distribution
5421:Statistical theory
5337:Probability theory
5217:Skorokhod integral
5187:Malliavin calculus
4770:Korn-Kreer-Lenssen
4654:Time series models
4617:Pitman–Yor process
4043:. pp. 57–58.
4041:Wiley-Interscience
3889:Santa Fe Institute
3810:
3580:
3552:
3504:
3480:
3029:
3021:
2998:
2981:
2958:
2920:
2912:
2889:
2872:
2849:
2826:
2825:
2804:
2803:
2782:
2781:
2760:
2755:
2739:
2723:
2704:
2697:
2690:
2662:
2657:
2641:
2622:
2615:
2587:
2582:
2566:
2547:
2540:
2512:
2507:
2491:
2472:
2465:
2437:
2436:
2415:
2414:
2393:
2392:
2368:
2367:
2346:
2345:
2324:
2313:
2285:
2274:
2243:
2238:
2219:
2207:
2179:
2178:
2157:
2156:
2135:
2134:
2113:
2112:
2091:
2086:
2074:
2046:
2035:
2007:
2006:
1985:
1984:
1960:
1955:
1931:
1903:
1898:
1891:
1863:
1858:
1842:
1823:
1799:
1771:
1770:
1749:
1748:
1719:
1673:
1498:
1496:
1153:
1107:
1065:
1063:
868:
848:
821:
706:
686:
663:
586:
566:
543:
455:
435:
411:
351:
291:
263:
243:
42:, a collection of
36:probability theory
32:
5403:
5402:
5357:Signal processing
5076:Doob's upcrossing
5071:Doob's martingale
5035:Engelbert–Schmidt
4978:Donsker's theorem
4912:Feller-continuous
4780:Rendleman–Bartter
4570:Dirichlet process
4487:Branching process
4456:Telegraph process
4349:Geometric process
4329:Empirical process
4319:Diffusion process
4175:Branching process
4170:Bernoulli process
4050:978-0-471-24195-9
3922:Statistics How To
3524:
3221:Bernoulli process
3091:Bernoulli process
3062:gambler's fallacy
1949:
1935:
1817:
1803:
1522:
1521:
1285:
1117:random variables
1110:{\displaystyle n}
1089:
1088:
871:{\displaystyle Y}
851:{\displaystyle X}
709:{\displaystyle Y}
689:{\displaystyle X}
589:{\displaystyle Y}
569:{\displaystyle X}
458:{\displaystyle Y}
438:{\displaystyle X}
266:{\displaystyle Y}
246:{\displaystyle X}
194:signal processing
82:signal processing
5428:
5377:Machine learning
5264:Usual hypotheses
5147:Girsanov theorem
5132:Dynkin's formula
4897:Continuous paths
4805:Actuarial models
4745:Garman–Kohlhagen
4715:Black–Karasinski
4710:Black–Derman–Toy
4697:Financial models
4563:Fields and other
4492:Gaussian process
4441:Sigma-martingale
4245:Additive process
4147:
4140:
4133:
4124:
4123:
4118:
4111:Berger, Roger L.
4092:
4091:
4089:
4088:
4073:
4067:
4061:
4055:
4054:
4032:
4026:
4025:
4015:
3991:
3985:
3983:
3966:
3938:
3932:
3931:
3929:
3928:
3913:
3907:
3906:
3904:
3903:
3897:
3891:. Archived from
3886:
3874:
3819:
3817:
3816:
3811:
3800:
3795:
3794:
3746:
3741:
3740:
3704:
3699:
3698:
3662:
3657:
3656:
3589:
3587:
3586:
3581:
3551:
3546:
3545:
3513:
3511:
3510:
3505:
3489:
3487:
3486:
3481:
3473:
3468:
3467:
3437:
3432:
3431:
3410:
3405:
3404:
3383:
3378:
3377:
3353:
3348:
3347:
3323:
3322:
3310:
3309:
3297:
3296:
3243:Machine learning
3173:Bruno de Finetti
3171:, introduced by
3038:
3036:
3035:
3030:
3028:
3027:
3022:
3005:
3004:
2999:
2988:
2987:
2982:
2967:
2965:
2964:
2959:
2929:
2927:
2926:
2921:
2919:
2918:
2913:
2896:
2895:
2890:
2879:
2878:
2873:
2858:
2856:
2855:
2850:
2835:
2833:
2832:
2827:
2813:
2811:
2810:
2805:
2791:
2789:
2788:
2783:
2769:
2767:
2766:
2761:
2756:
2740:
2724:
2705:
2698:
2691:
2671:
2669:
2668:
2663:
2658:
2642:
2623:
2616:
2596:
2594:
2593:
2588:
2583:
2567:
2548:
2541:
2521:
2519:
2518:
2513:
2508:
2492:
2473:
2466:
2446:
2444:
2443:
2438:
2424:
2422:
2421:
2416:
2402:
2400:
2399:
2394:
2377:
2375:
2374:
2369:
2355:
2353:
2352:
2347:
2333:
2331:
2330:
2325:
2314:
2294:
2292:
2291:
2286:
2275:
2252:
2250:
2249:
2244:
2239:
2220:
2213:
2208:
2188:
2186:
2185:
2180:
2166:
2164:
2163:
2158:
2144:
2142:
2141:
2136:
2122:
2120:
2119:
2114:
2100:
2098:
2097:
2092:
2087:
2080:
2075:
2055:
2053:
2052:
2047:
2036:
2016:
2014:
2013:
2008:
1994:
1992:
1991:
1986:
1969:
1967:
1966:
1961:
1956:
1947:
1946:
1933:
1932:
1912:
1910:
1909:
1904:
1899:
1892:
1872:
1870:
1869:
1864:
1859:
1843:
1824:
1815:
1814:
1801:
1800:
1780:
1778:
1777:
1772:
1758:
1756:
1755:
1750:
1728:
1726:
1725:
1720:
1718:
1717:
1699:
1698:
1682:
1680:
1679:
1674:
1669:
1668:
1656:
1655:
1637:
1636:
1624:
1623:
1599:
1598:
1580:
1579:
1567:
1566:
1565:
1564:
1546:
1545:
1516:
1507:
1505:
1504:
1499:
1497:
1487:
1486:
1468:
1467:
1449:
1448:
1436:
1435:
1434:
1433:
1407:
1406:
1394:
1393:
1392:
1391:
1371:
1370:
1352:
1351:
1339:
1338:
1337:
1336:
1318:
1317:
1302:
1286:
1283:
1239:
1238:
1237:
1236:
1210:
1209:
1208:
1207:
1192:
1180:
1162:
1160:
1159:
1154:
1152:
1151:
1133:
1132:
1116:
1114:
1113:
1108:
1083:
1074:
1072:
1071:
1066:
1064:
1030:
1029:
1008:
1007:
980:
979:
963:
935:
934:
913:
912:
902:
890:
877:
875:
874:
869:
857:
855:
854:
849:
830:
828:
827:
822:
792:
791:
770:
769:
742:
741:
715:
713:
712:
707:
695:
693:
692:
687:
672:
670:
669:
664:
640:
639:
618:
617:
595:
593:
592:
587:
575:
573:
572:
567:
552:
550:
549:
544:
491:
490:
464:
462:
461:
456:
444:
442:
441:
436:
420:
418:
417:
412:
377:
376:
360:
358:
357:
352:
317:
316:
300:
298:
297:
292:
290:
272:
270:
269:
264:
252:
250:
249:
244:
198:image processing
184:
44:random variables
5436:
5435:
5431:
5430:
5429:
5427:
5426:
5425:
5406:
5405:
5404:
5399:
5381:
5342:Queueing theory
5285:
5227:Skorokhod space
5090:
5081:Kunita–Watanabe
5052:
5018:Sanov's theorem
4988:Ergodic theorem
4961:
4957:Time-reversible
4875:
4838:Queueing models
4832:
4828:Sparre–Anderson
4818:Cramér–Lundberg
4799:
4785:SABR volatility
4691:
4648:
4600:Boolean network
4558:
4544:Renewal process
4475:
4424:Non-homogeneous
4414:Poisson process
4304:Contact process
4267:Brownian motion
4237:Continuous time
4231:
4225:Maximal entropy
4156:
4151:
4121:
4107:Casella, George
4101:
4099:Further reading
4096:
4095:
4086:
4084:
4075:
4074:
4070:
4062:
4058:
4051:
4033:
4029:
3992:
3988:
3955:10.2307/3315772
3939:
3935:
3926:
3924:
3914:
3910:
3901:
3899:
3895:
3884:
3875:
3871:
3866:
3844:
3796:
3790:
3786:
3742:
3736:
3732:
3700:
3694:
3690:
3658:
3652:
3648:
3604:
3601:
3600:
3547:
3526:
3525:
3522:
3519:
3518:
3499:
3496:
3495:
3469:
3463:
3459:
3433:
3427:
3423:
3406:
3400:
3396:
3379:
3373:
3369:
3349:
3343:
3339:
3318:
3314:
3305:
3301:
3292:
3288:
3265:
3262:
3261:
3240:
3206:
3200:
3185:symmetric group
3165:
3159:
3143:
3141:Generalizations
3121:
3099:
3070:
3050:
3045:
3023:
3016:
3015:
3000:
2993:
2992:
2983:
2976:
2975:
2973:
2970:
2969:
2935:
2932:
2931:
2914:
2907:
2906:
2891:
2884:
2883:
2874:
2867:
2866:
2864:
2861:
2860:
2844:
2841:
2840:
2819:
2816:
2815:
2797:
2794:
2793:
2775:
2772:
2771:
2750:
2734:
2718:
2699:
2692:
2685:
2677:
2674:
2673:
2652:
2636:
2617:
2610:
2602:
2599:
2598:
2577:
2561:
2542:
2535:
2527:
2524:
2523:
2502:
2486:
2467:
2460:
2452:
2449:
2448:
2430:
2427:
2426:
2408:
2405:
2404:
2386:
2383:
2382:
2361:
2358:
2357:
2339:
2336:
2335:
2308:
2300:
2297:
2296:
2269:
2261:
2258:
2257:
2233:
2214:
2209:
2202:
2194:
2191:
2190:
2172:
2169:
2168:
2150:
2147:
2146:
2128:
2125:
2124:
2106:
2103:
2102:
2081:
2076:
2069:
2061:
2058:
2057:
2030:
2022:
2019:
2018:
2000:
1997:
1996:
1978:
1975:
1974:
1950:
1936:
1926:
1918:
1915:
1914:
1893:
1886:
1878:
1875:
1874:
1853:
1837:
1818:
1804:
1794:
1786:
1783:
1782:
1764:
1761:
1760:
1742:
1739:
1738:
1735:
1713:
1709:
1694:
1690:
1688:
1685:
1684:
1664:
1660:
1651:
1647:
1632:
1628:
1619:
1615:
1594:
1590:
1575:
1571:
1560:
1556:
1541:
1537:
1536:
1532:
1530:
1527:
1526:
1523:
1495:
1494:
1482:
1478:
1463:
1459:
1454:
1444:
1440:
1429:
1425:
1424:
1420:
1402:
1398:
1387:
1383:
1382:
1378:
1366:
1362:
1347:
1343:
1332:
1328:
1313:
1309:
1308:
1304:
1300:
1299:
1284: and
1282:
1250:
1232:
1228:
1227:
1223:
1203:
1199:
1198:
1194:
1189:
1187:
1184:
1183:
1147:
1143:
1128:
1124:
1122:
1119:
1118:
1102:
1099:
1098:
1095:
1090:
1062:
1061:
1041:
1025:
1021:
1003:
999:
969:
965:
961:
960:
946:
930:
926:
908:
904:
899:
897:
894:
893:
863:
860:
859:
843:
840:
839:
831:. (See further
787:
783:
765:
761:
731:
727:
725:
722:
721:
720:if and only if
701:
698:
697:
681:
678:
677:
635:
631:
613:
609:
607:
604:
603:
581:
578:
577:
561:
558:
557:
480:
476:
474:
471:
470:
450:
447:
446:
430:
427:
426:
372:
368:
366:
363:
362:
312:
308:
306:
303:
302:
286:
278:
275:
274:
258:
255:
254:
238:
235:
234:
231:
226:
180:
177:Markov sequence
142:
90:
24:
17:
12:
11:
5:
5434:
5424:
5423:
5418:
5401:
5400:
5398:
5397:
5392:
5390:List of topics
5386:
5383:
5382:
5380:
5379:
5374:
5369:
5364:
5359:
5354:
5349:
5347:Renewal theory
5344:
5339:
5334:
5329:
5324:
5319:
5314:
5312:Ergodic theory
5309:
5304:
5302:Control theory
5299:
5293:
5291:
5287:
5286:
5284:
5283:
5282:
5281:
5276:
5266:
5261:
5256:
5251:
5246:
5245:
5244:
5234:
5232:Snell envelope
5229:
5224:
5219:
5214:
5209:
5204:
5199:
5194:
5189:
5184:
5179:
5174:
5169:
5164:
5159:
5154:
5149:
5144:
5139:
5134:
5129:
5124:
5119:
5114:
5109:
5104:
5098:
5096:
5092:
5091:
5089:
5088:
5083:
5078:
5073:
5068:
5062:
5060:
5054:
5053:
5051:
5050:
5031:Borel–Cantelli
5020:
5015:
5010:
5005:
5000:
4995:
4990:
4985:
4980:
4975:
4969:
4967:
4966:Limit theorems
4963:
4962:
4960:
4959:
4954:
4949:
4944:
4939:
4934:
4929:
4924:
4919:
4914:
4909:
4904:
4899:
4894:
4889:
4883:
4881:
4877:
4876:
4874:
4873:
4868:
4863:
4858:
4853:
4848:
4842:
4840:
4834:
4833:
4831:
4830:
4825:
4820:
4815:
4809:
4807:
4801:
4800:
4798:
4797:
4792:
4787:
4782:
4777:
4772:
4767:
4762:
4757:
4752:
4747:
4742:
4737:
4732:
4727:
4722:
4717:
4712:
4707:
4701:
4699:
4693:
4692:
4690:
4689:
4684:
4679:
4674:
4669:
4664:
4658:
4656:
4650:
4649:
4647:
4646:
4641:
4636:
4635:
4634:
4629:
4619:
4614:
4609:
4604:
4603:
4602:
4597:
4587:
4585:Hopfield model
4582:
4577:
4572:
4566:
4564:
4560:
4559:
4557:
4556:
4551:
4546:
4541:
4536:
4531:
4530:
4529:
4524:
4519:
4514:
4504:
4502:Markov process
4499:
4494:
4489:
4483:
4481:
4477:
4476:
4474:
4473:
4471:Wiener sausage
4468:
4466:Wiener process
4463:
4458:
4453:
4448:
4446:Stable process
4443:
4438:
4436:Semimartingale
4433:
4428:
4427:
4426:
4421:
4411:
4406:
4401:
4396:
4391:
4386:
4381:
4379:Jump diffusion
4376:
4371:
4366:
4361:
4356:
4354:Hawkes process
4351:
4346:
4341:
4336:
4334:Feller process
4331:
4326:
4321:
4316:
4311:
4306:
4301:
4299:Cauchy process
4296:
4295:
4294:
4289:
4284:
4279:
4274:
4264:
4263:
4262:
4252:
4250:Bessel process
4247:
4241:
4239:
4233:
4232:
4230:
4229:
4228:
4227:
4222:
4217:
4212:
4202:
4197:
4192:
4187:
4182:
4177:
4172:
4166:
4164:
4158:
4157:
4150:
4149:
4142:
4135:
4127:
4120:
4119:
4102:
4100:
4097:
4094:
4093:
4068:
4056:
4049:
4027:
3986:
3949:(3): 497–513,
3933:
3908:
3878:Clauset, Aaron
3868:
3867:
3865:
3862:
3861:
3860:
3855:
3850:
3843:
3840:
3839:
3838:
3834:
3833:distribution".
3822:
3821:
3809:
3806:
3803:
3799:
3793:
3789:
3785:
3782:
3779:
3776:
3773:
3770:
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3764:
3761:
3758:
3755:
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3739:
3735:
3731:
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3725:
3722:
3719:
3716:
3713:
3710:
3707:
3703:
3697:
3693:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3665:
3661:
3655:
3651:
3647:
3644:
3641:
3638:
3635:
3632:
3629:
3626:
3623:
3620:
3617:
3614:
3611:
3608:
3592:
3591:
3579:
3576:
3573:
3570:
3567:
3564:
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3558:
3555:
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3413:
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3372:
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3342:
3338:
3335:
3332:
3329:
3326:
3321:
3317:
3313:
3308:
3304:
3300:
3295:
3291:
3287:
3284:
3281:
3278:
3275:
3272:
3269:
3239:
3236:
3232:Wiener process
3228:LĂ©vy processes
3202:Main article:
3199:
3196:
3161:Main article:
3158:
3155:
3151:distributional
3142:
3139:
3138:
3137:
3131:
3120:
3117:
3116:
3115:
3109:
3098:
3095:
3087:
3086:
3080:
3069:
3066:
3049:
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3041:
3026:
3020:
3014:
3011:
3008:
3003:
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2986:
2980:
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2911:
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2877:
2871:
2852:{\textstyle n}
2848:
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2111:
2090:
2085:
2079:
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2068:
2065:
2045:
2042:
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2034:
2029:
2026:
2005:
1983:
1959:
1954:
1945:
1942:
1939:
1930:
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1605:
1602:
1597:
1593:
1589:
1586:
1583:
1578:
1574:
1570:
1563:
1559:
1555:
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1544:
1540:
1535:
1520:
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1510:
1508:
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1471:
1466:
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1397:
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1278:
1275:
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1257:
1254:
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1020:
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995:
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986:
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975:
972:
968:
964:
962:
959:
956:
953:
950:
947:
944:
941:
938:
933:
929:
925:
922:
919:
916:
911:
907:
903:
901:
888:
867:
847:
820:
817:
814:
811:
808:
805:
801:
798:
795:
790:
786:
782:
779:
776:
773:
768:
764:
760:
757:
754:
751:
748:
745:
740:
737:
734:
730:
705:
685:
662:
659:
656:
653:
649:
646:
643:
638:
634:
630:
627:
624:
621:
616:
612:
601:if and only if
585:
565:
542:
539:
536:
533:
530:
527:
524:
521:
518:
515:
512:
509:
506:
503:
500:
497:
494:
489:
486:
483:
479:
454:
434:
410:
407:
404:
401:
398:
395:
392:
389:
386:
383:
380:
375:
371:
350:
347:
344:
341:
338:
335:
332:
329:
326:
323:
320:
315:
311:
289:
285:
282:
262:
242:
230:
227:
225:
222:
141:
138:
137:
136:
129:
89:
86:
15:
9:
6:
4:
3:
2:
5433:
5422:
5419:
5417:
5414:
5413:
5411:
5396:
5393:
5391:
5388:
5387:
5384:
5378:
5375:
5373:
5370:
5368:
5365:
5363:
5360:
5358:
5355:
5353:
5350:
5348:
5345:
5343:
5340:
5338:
5335:
5333:
5330:
5328:
5325:
5323:
5320:
5318:
5315:
5313:
5310:
5308:
5305:
5303:
5300:
5298:
5295:
5294:
5292:
5288:
5280:
5277:
5275:
5272:
5271:
5270:
5267:
5265:
5262:
5260:
5257:
5255:
5252:
5250:
5249:Stopping time
5247:
5243:
5240:
5239:
5238:
5235:
5233:
5230:
5228:
5225:
5223:
5220:
5218:
5215:
5213:
5210:
5208:
5205:
5203:
5200:
5198:
5195:
5193:
5190:
5188:
5185:
5183:
5180:
5178:
5175:
5173:
5170:
5168:
5165:
5163:
5160:
5158:
5155:
5153:
5150:
5148:
5145:
5143:
5140:
5138:
5135:
5133:
5130:
5128:
5125:
5123:
5120:
5118:
5115:
5113:
5110:
5108:
5105:
5103:
5100:
5099:
5097:
5093:
5087:
5084:
5082:
5079:
5077:
5074:
5072:
5069:
5067:
5064:
5063:
5061:
5059:
5055:
5048:
5044:
5040:
5039:Hewitt–Savage
5036:
5032:
5028:
5024:
5023:Zero–one laws
5021:
5019:
5016:
5014:
5011:
5009:
5006:
5004:
5001:
4999:
4996:
4994:
4991:
4989:
4986:
4984:
4981:
4979:
4976:
4974:
4971:
4970:
4968:
4964:
4958:
4955:
4953:
4950:
4948:
4945:
4943:
4940:
4938:
4935:
4933:
4930:
4928:
4925:
4923:
4920:
4918:
4915:
4913:
4910:
4908:
4905:
4903:
4900:
4898:
4895:
4893:
4890:
4888:
4885:
4884:
4882:
4878:
4872:
4869:
4867:
4864:
4862:
4859:
4857:
4854:
4852:
4849:
4847:
4844:
4843:
4841:
4839:
4835:
4829:
4826:
4824:
4821:
4819:
4816:
4814:
4811:
4810:
4808:
4806:
4802:
4796:
4793:
4791:
4788:
4786:
4783:
4781:
4778:
4776:
4773:
4771:
4768:
4766:
4763:
4761:
4758:
4756:
4753:
4751:
4748:
4746:
4743:
4741:
4738:
4736:
4733:
4731:
4728:
4726:
4723:
4721:
4720:Black–Scholes
4718:
4716:
4713:
4711:
4708:
4706:
4703:
4702:
4700:
4698:
4694:
4688:
4685:
4683:
4680:
4678:
4675:
4673:
4670:
4668:
4665:
4663:
4660:
4659:
4657:
4655:
4651:
4645:
4642:
4640:
4637:
4633:
4630:
4628:
4625:
4624:
4623:
4622:Point process
4620:
4618:
4615:
4613:
4610:
4608:
4605:
4601:
4598:
4596:
4593:
4592:
4591:
4588:
4586:
4583:
4581:
4580:Gibbs measure
4578:
4576:
4573:
4571:
4568:
4567:
4565:
4561:
4555:
4552:
4550:
4547:
4545:
4542:
4540:
4537:
4535:
4532:
4528:
4525:
4523:
4520:
4518:
4515:
4513:
4510:
4509:
4508:
4505:
4503:
4500:
4498:
4495:
4493:
4490:
4488:
4485:
4484:
4482:
4478:
4472:
4469:
4467:
4464:
4462:
4459:
4457:
4454:
4452:
4449:
4447:
4444:
4442:
4439:
4437:
4434:
4432:
4429:
4425:
4422:
4420:
4417:
4416:
4415:
4412:
4410:
4407:
4405:
4402:
4400:
4397:
4395:
4392:
4390:
4387:
4385:
4382:
4380:
4377:
4375:
4372:
4370:
4369:ItĂ´ diffusion
4367:
4365:
4362:
4360:
4357:
4355:
4352:
4350:
4347:
4345:
4344:Gamma process
4342:
4340:
4337:
4335:
4332:
4330:
4327:
4325:
4322:
4320:
4317:
4315:
4312:
4310:
4307:
4305:
4302:
4300:
4297:
4293:
4290:
4288:
4285:
4283:
4280:
4278:
4275:
4273:
4270:
4269:
4268:
4265:
4261:
4258:
4257:
4256:
4253:
4251:
4248:
4246:
4243:
4242:
4240:
4238:
4234:
4226:
4223:
4221:
4218:
4216:
4215:Self-avoiding
4213:
4211:
4208:
4207:
4206:
4203:
4201:
4200:Moran process
4198:
4196:
4193:
4191:
4188:
4186:
4183:
4181:
4178:
4176:
4173:
4171:
4168:
4167:
4165:
4163:
4162:Discrete time
4159:
4155:
4148:
4143:
4141:
4136:
4134:
4129:
4128:
4125:
4116:
4112:
4108:
4104:
4103:
4082:
4078:
4072:
4065:
4060:
4052:
4046:
4042:
4038:
4031:
4023:
4019:
4014:
4009:
4005:
4001:
3997:
3990:
3982:
3978:
3974:
3970:
3965:
3960:
3956:
3952:
3948:
3944:
3937:
3923:
3919:
3912:
3898:on 2012-01-20
3894:
3890:
3883:
3879:
3873:
3869:
3859:
3856:
3854:
3851:
3849:
3846:
3845:
3835:
3831:
3830:
3829:
3826:
3801:
3791:
3787:
3780:
3774:
3771:
3768:
3765:
3762:
3759:
3756:
3747:
3737:
3733:
3726:
3720:
3717:
3714:
3705:
3695:
3691:
3684:
3678:
3675:
3672:
3663:
3653:
3649:
3642:
3636:
3633:
3630:
3621:
3615:
3609:
3606:
3599:
3598:
3597:
3596:
3571:
3565:
3559:
3556:
3553:
3548:
3517:
3516:
3515:
3501:
3474:
3464:
3460:
3453:
3450:
3447:
3444:
3438:
3428:
3424:
3417:
3411:
3401:
3397:
3390:
3384:
3374:
3370:
3363:
3360:
3354:
3344:
3340:
3336:
3333:
3330:
3327:
3324:
3319:
3315:
3311:
3306:
3302:
3298:
3293:
3289:
3282:
3279:
3273:
3267:
3260:
3259:
3258:
3254:
3252:
3247:
3244:
3235:
3233:
3229:
3224:
3222:
3218:
3215:
3214:discrete time
3211:
3205:
3195:
3193:
3188:
3186:
3182:
3178:
3174:
3170:
3164:
3154:
3152:
3148:
3135:
3132:
3129:
3126:
3125:
3124:
3113:
3110:
3107:
3104:
3103:
3102:
3094:
3092:
3084:
3081:
3078:
3075:
3074:
3073:
3065:
3063:
3059:
3055:
3040:
3024:
3018:
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2003:
1981:
1971:
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1913:is short for
1895:
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309:
283:
280:
260:
240:
221:
219:
215:
214:deconvolution
211:
207:
205:
201:
199:
195:
190:
188:
183:
178:
173:
168:
166:
163:approaches a
162:
158:
154:
149:
147:
133:
130:
128:distribution.
127:
123:
120:
119:
118:
116:
112:
111:random sample
108:
107:
102:
101:random sample
97:
95:
85:
83:
79:
75:
74:
69:
68:
63:
62:
57:
53:
49:
45:
41:
37:
28:
22:
5307:Econometrics
5269:Wiener space
5157:ItĂ´ integral
5058:Inequalities
4947:Self-similar
4917:Gauss–Markov
4907:Exchangeable
4887:CĂ dlĂ g paths
4823:Risk process
4775:LIBOR market
4644:Random graph
4639:Random field
4451:Superprocess
4389:LĂ©vy process
4384:Jump process
4359:Hunt process
4195:Markov chain
4189:
4114:
4085:. Retrieved
4083:. 2020-05-05
4080:
4071:
4059:
4036:
4030:
4003:
3999:
3989:
3946:
3942:
3936:
3925:. Retrieved
3921:
3911:
3900:. Retrieved
3893:the original
3872:
3827:
3823:
3594:
3593:
3493:
3255:
3250:
3248:
3241:
3227:
3225:
3217:LĂ©vy process
3207:
3204:LĂ©vy process
3198:LĂ©vy process
3189:
3166:
3153:assumption.
3146:
3144:
3133:
3127:
3122:
3111:
3105:
3100:
3088:
3082:
3076:
3071:
3057:
3051:
2838:
2380:
2255:
1972:
1736:
1524:
1512:
1172:
1164:
1096:
1079:
883:
879:
837:
717:
675:
597:
555:
232:
209:
208:
203:
202:
191:
187:sample space
181:
171:
169:
152:
150:
143:
131:
121:
114:
110:
105:
104:
100:
98:
93:
91:
88:Introduction
72:
71:
66:
65:
60:
59:
47:
33:
5352:Ruin theory
5290:Disciplines
5162:ItĂ´'s lemma
4937:Predictable
4612:Percolation
4595:Potts model
4590:Ising model
4554:White noise
4512:Differences
4374:ItĂ´ process
4314:Cox process
4210:Loop-erased
4205:Random walk
4006:: 222–229.
3177:permutation
3128:Independent
3106:Independent
3077:Independent
2189:, there is
718:independent
218:white noise
140:Application
132:Independent
126:probability
78:data mining
56:independent
5410:Categories
5362:Statistics
5142:Filtration
5043:Kolmogorov
5027:Blumenthal
4952:Stationary
4892:Continuous
4880:Properties
4765:Hull–White
4507:Martingale
4394:Local time
4282:Fractional
4260:pure birth
4087:2021-12-16
3927:2021-12-09
3902:2011-11-29
3864:References
224:Definition
40:statistics
5274:Classical
4287:Geometric
4277:Excursion
4081:Expert.ai
4022:124843240
3802:θ
3775:
3748:θ
3721:
3706:θ
3679:
3664:θ
3637:
3622:θ
3610:
3572:θ
3560:
3554:
3549:θ
3502:θ
3475:θ
3439:θ
3412:θ
3385:θ
3355:θ
3274:θ
3119:Example 4
3097:Example 3
3068:Example 2
3056:wheel is
3048:Example 1
3010:…
2950:…
2901:…
2256:Note: If
1704:…
1658:≤
1645:∧
1642:…
1639:∧
1626:≤
1610:
1585:…
1551:…
1489:∈
1473:…
1457:∀
1418:⋅
1415:…
1412:⋅
1357:…
1323:…
1294:∈
1288:∀
1271:…
1259:∈
1253:∀
1178:EQUATION
1138:…
1056:∈
1044:∀
1019:⋅
955:∈
949:∀
816:∈
804:∀
781:⋅
658:∈
652:∀
535:≤
529:∧
523:≤
514:
403:≤
394:
343:≤
334:
284:⊆
5395:Category
5279:Abstract
4813:BĂĽhlmann
4419:Compound
4113:(2002),
3981:53117661
3880:(2011).
3842:See also
3054:roulette
3043:Examples
2859:events,
2381:Suppose
161:variance
4902:Ergodic
4790:VašĂÄŤek
4632:Poisson
4292:Meander
3973:3315772
2334:, then
421:be the
216:) to a
5242:Tanaka
4927:Mixing
4922:Markov
4795:Wilkie
4760:Ho–Lee
4755:Heston
4527:Super-
4272:Bridge
4220:Biased
4047:
4020:
3979:
3971:
3251:i.i.d.
2814:, and
2672:, and
2425:, and
1948:
1934:
1816:
1802:
1525:where
1165:i.i.d.
880:i.i.d.
301:. Let
172:i.i.d.
153:i.i.d.
135:versa.
61:i.i.d.
5095:Tools
4871:M/M/c
4866:M/M/1
4861:M/G/1
4851:Fluid
4517:Local
4018:S2CID
3984:(§8).
3977:S2CID
3969:JSTOR
3896:(PDF)
3885:(PDF)
3595:where
3147:i.i.d
3058:i.i.d
2017:. If
70:, or
5047:LĂ©vy
4846:Bulk
4730:Chen
4522:Sub-
4480:Both
4045:ISBN
3249:The
2356:and
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2295:and
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2041:>
1995:and
1759:and
1515:Eq.2
1163:are
1082:Eq.1
878:are
858:and
716:are
696:and
596:are
576:and
445:and
361:and
253:and
196:and
170:The
151:The
103:and
80:and
38:and
4627:Cox
4008:doi
3959:hdl
3951:doi
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1173:and
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469:by
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