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Moved page from wikipedia:en:Stochastic process (history)

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===Classifications===

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The definition of a stochastic process varies, but a stochastic process is traditionally defined as a collection of random variables indexed by some set. Both "collection", while instead of "index set", sometimes the terms "parameter set" though sometimes it is only used when the stochastic process takes real values. while the terms stochastic process and random process are usually used when the index set is interpreted as time, and other terms are used such as random field when the index set is n-dimensional Euclidean space \mathbb{R}^n or a manifold. \{X(t)\} or simply as X or X(t), although X(t) is regarded as an abuse of function notation. For example, X(t) or X_t are used to refer to the random variable with the index t, and not the entire stochastic process. In other words, a Bernoulli process is a sequence of iid Bernoulli random variables, where each coin flip is an example of a Bernoulli trial.

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The definition of a stochastic process varies, but a stochastic process is traditionally defined as a collection of random variables indexed by some set. Both "collection", while instead of "index set", sometimes the terms "parameter set" though sometimes it is only used when the stochastic process takes real values. while the terms stochastic process and random process are usually used when the index set is interpreted as time, and other terms are used such as random field when the index set is <math>n</math>-dimensional Euclidean space <math>\mathbb{R}^n</math> or a manifold. <math>\{X(t)\}</math> or simply as <math>X</math> or <math>X(t)</math>, although <math>X(t)</math> is regarded as an abuse of function notation. For example, <math>X(t)</math> or <math>X_t</math> are used to refer to the random variable with the index <math>t</math>, and not the entire stochastic process. In other words, a Bernoulli process is a sequence of iid Bernoulli random variables, where each coin flip is an example of a Bernoulli trial.

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随机过程的定义各不相同，但随机过程通常被定义为由一组随机变量组成的集合。两者都是“集合” ，而不是“索引集合” ，有时使用术语“参数集合” ~~，但有时只有在随机过程数据库采用真实值时才使用。当索引集被解释为时间时，通常使用随机过程和随机过程，当索引集是~~ n ~~维欧氏空间~~ mathbb { r } ^ n ~~或流形时，则使用随机场等其他术语。{ x (t)}或简单地作为 x 或~~ x (t) ~~，尽管 x (t)被认为是滥用函数表示法。例如，x (t)~~或 x _ t ~~用于引用索引为~~ t 的随机变量，而不是整个随机过程。换句话说，伯努利过程是一系列 iid Bernoulli 随机变量，每次抛硬币都是 Bernoulli 试验的一个例子。

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随机过程的定义各不相同，但随机过程通常被定义为由一组随机变量组成的集合。两者都是“集合” ，而不是“索引集合” ，有时使用术语“参数集合” ，但有时只有在随机过程数据库采用真实值时才使用。当索引集被解释为时间时，通常使用术语随机过程和随机过程，当索引集是 < math > n </math >-dimensional Euclidean space < math > mathbb { r } ^ n </math > 或者是流形时，则使用随机场。虽然《 math 》被认为是对函数表示法的滥用，但《 math 》还是被简单地称为《 math 》或《 math 》。例如，< math > x (t) </math > 或 < math > x _ t </math > 用于指代带有索引 < math > t </math > 的随机变量，而不是整个随机过程。换句话说，伯努利过程是一系列 iid Bernoulli 随机变量，每次抛硬币都是 Bernoulli 试验的一个例子。

A stochastic process can be classified in different ways, for example, by its state space, its index set, or the dependence among the random variables. One common way of classification is by the [[cardinality]] of the index set and the state space.<ref name="Florescu2014page294"/><ref name="KarlinTaylor2012page26">{{cite book|author1=Samuel Karlin|author2=Howard E. Taylor|title=A First Course in Stochastic Processes|url=https://books.google.com/books?id=dSDxjX9nmmMC|year=2012|publisher=Academic Press|isbn=978-0-08-057041-9|page=26}}</ref><ref>{{cite book|author1=Donald L. Snyder|author2=Michael I. Miller|title=Random Point Processes in Time and Space|url=https://books.google.com/books?id=c_3UBwAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4612-3166-0|pages=24, 25}}</ref>

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===Etymology===

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A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, p, or decreases by one with probability 1-p, so the index set of this random walk is the natural numbers, while its state space is the integers. If the p=0.5, this random walk is called a symmetric random walk.

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A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, <math>p</math>, or decreases by one with probability <math>1-p</math>, so the index set of this random walk is the natural numbers, while its state space is the integers. If the <math>p=0.5</math>, this random walk is called a symmetric random walk.

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一个经典的随机游走的例子被称为简单随机游走，这是一个以整数为状态空间的离散时间随机过程，它基于一个伯努利过程，其中每个 Bernoulli 变量要么取值为正，要么取值为负。换句话说，简单随机游动发生在整数上，它的值随概率的增加而增加一倍，如 p，或者随概率的减少而减少一倍，因此这种随机游动的指数集是自然数，而它的状态空间是整数。如果 p = 0.~~5，这种随机游动称为对称随机游动。~~

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一个经典的随机游走的例子被称为简单随机游走，这是一个以整数为状态空间的离散时间随机过程，它基于一个伯努利过程，其中每个 Bernoulli 变量要么取值为正，要么取值为负。换句话说，简单随机游动发生在整数上，它的值随概率的增加而增加1，或随概率的减少而减少1，所以这种随机游动的指数集是自然数，而它的状态空间是整数。如果 < math > p = 0.5 </math > ，这种随机漫步称为对称随机漫步。

The word ''stochastic'' in [[English language|English]] was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a [[Greek language|Greek]] word meaning "to aim at a mark, guess", and the [[Oxford English Dictionary]] gives the year 1662 as its earliest occurrence.<ref name="OxfordStochastic">{{Cite OED|Stochastic}}</ref> In his work on probability ''Ars Conjectandi'', originally published in Latin in 1713, [[Jakob Bernoulli]] used the phrase "Ars Conjectandi sive Stochastice", which has been translated to "the art of conjecturing or stochastics".<ref name="Sheĭnin2006page5">{{cite book|author=O. B. Sheĭnin|title=Theory of probability and statistics as exemplified in short dictums|url=https://books.google.com/books?id=XqMZAQAAIAAJ|year=2006|publisher=NG Verlag|isbn=978-3-938417-40-9|page=5}}</ref> This phrase was used, with reference to Bernoulli, by [[Ladislaus Bortkiewicz]]<ref name="SheyninStrecker2011page136">{{cite book|author1=Oscar Sheynin|author2=Heinrich Strecker|title=Alexandr A. Chuprov: Life, Work, Correspondence|url=https://books.google.com/books?id=1EJZqFIGxBIC&pg=PA9|year=2011|publisher=V&R unipress GmbH|isbn=978-3-89971-812-6|page=136}}</ref> who in 1917 wrote in German the word ''stochastik'' with a sense meaning random. The term ''stochastic process'' first appeared in English in a 1934 paper by [[Joseph Doob]].<ref name="OxfordStochastic"/> For the term and a specific mathematical definition, Doob cited another 1934 paper, where the term ''stochastischer Prozeß'' was used in German by [[Aleksandr Khinchin]],<ref name="Doob1934"/><ref name="Khintchine1934">{{cite journal|last1=Khintchine|first1=A.|title=Korrelationstheorie der stationeren stochastischen Prozesse|journal=Mathematische Annalen|volume=109|issue=1|year=1934|pages=604–615|issn=0025-5831|doi=10.1007/BF01449156}}</ref> though the German term had been used earlier, for example, by Andrei Kolmogorov in 1931.<ref name="Kolmogoroff1931page1">{{cite journal|last1=Kolmogoroff|first1=A.|title=Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung|journal=Mathematische Annalen|volume=104|issue=1|year=1931|page=1|issn=0025-5831|doi=10.1007/BF01457949}}</ref>

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Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes. Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space. But the process can be defined more generally so its state space can be n-dimensional Euclidean space. If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant \mu, which is a real number, then the resulting stochastic process is said to have drift \mu.

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Playing a central role in the theory of probability, the Wiener process is often considered the most important and studied stochastic process, with connections to other stochastic processes. Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space. But the process can be defined more generally so its state space can be <math>n</math>-dimensional Euclidean space. If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant <math> \mu</math>, which is a real number, then the resulting stochastic process is said to have drift <math> \mu</math>.

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在概率论中起着核心作用的维纳过程，通常被认为是最重要的和研究过的随机过程过程，与其他随机过程有联系。它的索引集和状态空间分别为非负数和实数，因此它既有连续索引集又有状态空间。但是这个过程可以定义得更广泛，因此它的状态空间可以是 n 维欧氏空间。如果任何增量的平均值为零，那么由此产生的 Wiener 或 Brownian ~~运动过程称为零漂过程。如果任意两个时间点的增量的平均值等于时间差乘以某个常数 μ，这是一个实数，那么得到的随机过程就是漂移 μ。~~

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在概率论中起着核心作用的维纳过程，通常被认为是最重要的和研究过的随机过程过程，与其他随机过程有联系。它的索引集和状态空间分别为非负数和实数，因此它既有连续索引集又有状态空间。但是这个过程可以定义得更广泛，因此它的状态空间可以是维的欧氏空间。如果任何增量的平均值为零，那么由此产生的 Wiener 或 Brownian 运动过程称为零漂过程。如果任意两个时间点的增量的平均值等于时间差乘以某个常数，即一个实数，那么得到的随机过程就是漂移。

===Notation===

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One of the simplest stochastic processes is the [[Bernoulli process]],<ref name="Florescu2014page293"/> which is a sequence of [[independent and identically distributed]] (iid) random variables, where each random variable takes either the value one or zero, say one with probability <math>p</math> and zero with probability <math>1-p</math>. This process can be linked to repeatedly flipping a coin, where the probability of obtaining a head is <math>p</math> and its value is one, while the value of a tail is zero.<ref name="Florescu2014page301">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|page=301}}</ref> In other words, a Bernoulli process is a sequence of [[Independent and identically distributed random variables|iid]] Bernoulli random variables,<ref name="BertsekasTsitsiklis2002page273">{{cite book|author1=Dimitri P. Bertsekas|author2=John N. Tsitsiklis|title=Introduction to Probability|url=https://books.google.com/books?id=bcHaAAAAMAAJ|year=2002|publisher=Athena Scientific|isbn=978-1-886529-40-3|page=273}}</ref> where each coin flip is an example of a [[Bernoulli trial]].<ref name="Ibe2013page11">{{cite book|author=Oliver C. Ibe|title=Elements of Random Walk and Diffusion Processes|url=https://books.google.com/books?id=DUqaAAAAQBAJ&pg=PT10|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-61793-9|page=11}}</ref>

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The Poisson process is a stochastic process that has different forms and definitions. It can be defined as a counting process, which is a stochastic process that represents the random number of points or events up to some time. The number of points of the process that are located in the interval from zero to some given time is a Poisson random variable that depends on that time and some parameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process. The homogeneous Poisson process is a member of important classes of stochastic processes such as Markov processes and Lévy processes. If the parameter constant of the Poisson process is replaced with some non-negative integrable function of t, the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randomly occurring in certain time windows.

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The Poisson process is a stochastic process that has different forms and definitions. It can be defined as a counting process, which is a stochastic process that represents the random number of points or events up to some time. The number of points of the process that are located in the interval from zero to some given time is a Poisson random variable that depends on that time and some parameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process. The homogeneous Poisson process is a member of important classes of stochastic processes such as Markov processes and Lévy processes. If the parameter constant of the Poisson process is replaced with some non-negative integrable function of <math>t</math>, the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant. Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randomly occurring in certain time windows.

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泊松过程是一个具有不同形式和定义的随机过程过程。它可以被定义为一个计数过程，这是一个随机过程，代表点或事件的随机数到一定时间。从零到给定时间区间内的过程点数是泊松随机变量，取决于该时间和某些参数。该过程以自然数为状态空间，非负数为索引集。这个过程也被称为泊松计数过程，因为它可以被解释为计数过程的一个例子。齐次泊松过程是一类重要的随机过程，如马尔可夫过程和 Lévy 过程的成员。如果将泊松过程的参数常数替换为 t 的某个非负可积函数，得到的过程称为非齐次或非齐次泊松过程，其点的平均密度不再是常数。泊松过程作为排队论中的一个基本过程，是数学模型中的一个重要过程，它在特定时间窗内随机发生的事件模型中找到了应用。

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泊松过程是一个具有不同形式和定义的随机过程过程。它可以被定义为一个计数过程，这是一个随机过程，代表点或事件的随机数到一定时间。从零到给定时间区间内的过程点数是泊松随机变量，取决于该时间和某些参数。该过程以自然数为状态空间，非负数为索引集。这个过程也被称为泊松计数过程，因为它可以被解释为计数过程的一个例子。齐次泊松过程是一类重要的随机过程，如马尔可夫过程和 Lévy 过程的成员。如果将泊松过程的参数常数替换为 < math > t </math > 的非负可积函数，则得到的过程称为非齐次或非齐次泊松过程，其点的平均密度不再是常数。泊松过程作为排队论中的一个基本过程，是数学模型中的一个重要过程，它在特定时间窗内随机发生的事件模型中找到了应用。

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===Random walk===

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Defined on the real line, the Poisson process can be interpreted as a stochastic process, among other random objects. But then it can be defined on the n-dimensional Euclidean space or other mathematical spaces, where it is often interpreted as a random set or a random counting measure, instead of a stochastic process. But it has been remarked that the Poisson process does not receive as much attention as it should, partly due to it often being considered just on the real line, and not on other mathematical spaces.

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Defined on the real line, the Poisson process can be interpreted as a stochastic process, among other random objects. But then it can be defined on the <math>n</math>-dimensional Euclidean space or other mathematical spaces, where it is often interpreted as a random set or a random counting measure, instead of a stochastic process. But it has been remarked that the Poisson process does not receive as much attention as it should, partly due to it often being considered just on the real line, and not on other mathematical spaces.

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在实际线上定义的泊松过程可以被解释为随机过程过程，以及其他随机对象。但是，它可以定义在 n 维欧氏空间或其他数学空间，在那里，它经常被解释为一个随机集或随机计数测度，而不是一个随机过程。但是人们注意到泊松过程并没有得到应有的重视，部分原因是泊松过程通常只考虑实线，而不考虑其他数学空间。

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在实际线上定义的泊松过程可以被解释为随机过程过程，以及其他随机对象。但是，它可以定义在维欧氏空间或其他数学空间上，在这些空间中，它通常被解释为一个随机集或随机计数测度，而不是一个随机过程。但是人们注意到泊松过程并没有得到应有的重视，部分原因是泊松过程通常只考虑实线，而不考虑其他数学空间。

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A classic example of a random walk is known as the ''simple random walk'', which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one. In other words, the simple random walk takes place on the integers, and its value increases by one with probability, say, <math>p</math>, or decreases by one with probability <math>1-p</math>, so the index set of this random walk is the natural numbers, while its state space is the integers. If the <math>p=0.5</math>, this random walk is called a symmetric random walk.<ref name="Gut2012page88">{{cite book|author=Allan Gut|title=Probability: A Graduate Course|url=https://books.google.com/books?id=XDFA-n_M5hMC|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4614-4708-5|page=88}}</ref><ref name="GrimmettStirzaker2001page71">{{cite book|author1=Geoffrey Grimmett|author2=David Stirzaker|title=Probability and Random Processes|url=https://books.google.com/books?id=G3ig-0M4wSIC|year=2001|publisher=OUP Oxford|isbn=978-0-19-857222-0|page=71}}</ref>

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A stochastic process is defined as a collection of random variables defined on a common probability space (\Omega, \mathcal{F}, P), where \Omega is a sample space, \mathcal{F} is a \sigma-algebra, and P is a probability measure; and the random variables, indexed by some set T, all take values in the same mathematical space S, which must be measurable with respect to some \sigma-algebra \Sigma.

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A stochastic process is defined as a collection of random variables defined on a common probability space <math>(\Omega, \mathcal{F}, P)</math>, where <math>\Omega</math> is a sample space, <math>\mathcal{F}</math> is a <math>\sigma</math>-algebra, and <math>P</math> is a probability measure; and the random variables, indexed by some set <math>T</math>, all take values in the same mathematical space <math>S</math>, which must be measurable with respect to some <math>\sigma</math>-algebra <math>\Sigma</math>.

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~~一个随机过程是定义在一个公共的概率空间上的随机变量的集合~~(Omega，mathcal { f } ，p) ，其中 Omega ~~是一个样本空间，mathcal~~ { f }~~是一个 Sigma~~-~~algebra，p 是一个 Sigma-algebra~~; ~~而随机变量，由一些集合 t 索引，都在相同的数学空间 s 中取值，这些值对于一些 Sigma-Sigma 代数必须是可测量的。~~

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随机过程被定义为一系列随机变量的集合，这些随机变量定义在一个普通的概率空间上(Omega，mathcal { f } ，p) </math > ，其中 < math > Omega </math > 是一个样本空间，< math > mathcal { f } </math > 是 < math > sigma </math >-algebra，而 < math > p </math > 是一个机率量测;以及随机变量，用一些集合作为指标，它们都在同一个数学空间中取值，这些值必须是可以测量的。

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< 中心 > < 数学 >

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< 中心 > < 数学 >

===Wiener process===

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The Wiener process is a stochastic process with stationary and [[independent increments]] that are [[normally distributed]] based on the size of the increments.<ref name="RogersWilliams2000page1">{{cite book|author1=L. C. G. Rogers|author2=David Williams|title=Diffusions, Markov Processes, and Martingales: Volume 1, Foundations|url=https://books.google.com/books?id=W0ydAgAAQBAJ&pg=PA1|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|page=1}}</ref><ref name="Klebaner2005page56">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|page=56}}</ref> The Wiener process is named after [[Norbert Wiener]], who proved its mathematical existence, but the process is also called the Brownian motion process or just Brownian motion due to its historical connection as a model for [[Brownian movement]] in liquids.<ref name="Brush1968page1">{{cite journal|last1=Brush|first1=Stephen G.|title=A history of random processes|journal=Archive for History of Exact Sciences|volume=5|issue=1|year=1968|pages=1–2|issn=0003-9519|doi=10.1007/BF00328110}}</ref><ref name="Applebaum2004page1338">{{cite journal|last1=Applebaum|first1=David|title=Lévy processes: From probability to finance and quantum groups|journal=Notices of the AMS|volume=51|issue=11|year=2004|pages=1338}}</ref><ref name="Applebaum2004page1338"/><ref name="GikhmanSkorokhod1969page21">{{cite book|author1=Iosif Ilyich Gikhman|author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=yJyLzG7N7r8C&pg=PR2|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=21}}</ref>

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Historically, in many problems from the natural sciences a point t\in T had the meaning of time, so X(t) is a random variable representing a value observed at time t. A stochastic process can also be written as \{X(t,\omega):t\in T \} to reflect that it is actually a function of two variables, t\in T and \omega\in \Omega.

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Historically, in many problems from the natural sciences a point <math>t\in T</math> had the meaning of time, so <math>X(t)</math> is a random variable representing a value observed at time <math>t</math>. A stochastic process can also be written as <math> \{X(t,\omega):t\in T \}</math> to reflect that it is actually a function of two variables, <math>t\in T</math> and <math>\omega\in \Omega</math>.

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从历史上看，在自然科学的许多问题中，t ~~中的一个点 t 具有时间的意义，因此~~ x (t)~~是一个随机变量，代表在时间 t 观测到的一个值。随机过程也可以写成 t 中的~~{ x (~~t，ω~~) : ~~t，以反映它实际上是一个双变量的函数，t~~ 中的 t 和 ω 中的 ω。

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从历史上看，在自然科学的许多问题中，t 中的一个点“ math”代表时间，所以“ math” x (t)是一个随机变量，代表时间观察到的值。一个随机过程也可以写成{ math > { x (t，Omega) : t in t } </math > 来反映它实际上是一个双变量的函数，t </math > 中的 < math > 和 ω </math > 中的 < math > Omega。

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[[File:DriftedWienerProcess1D.svg|thumb|left|Realizations of Wiener processes (or Brownian motion processes) with drift ({{color|blue|blue}}) and without drift ({{color|red|red}}).]]

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There are other ways to consider a stochastic process, with the above definition being considered the traditional one. For example, a stochastic process can be interpreted or defined as a S^T-valued random variable, where S^T is the space of all the possible S-valued functions of t\in T that map from the set T into the space S. of the stochastic process. Often this set is some subset of the real line, such as the natural numbers or an interval, giving the set T the interpretation of time. such as the Cartesian plane R^2 or n-dimensional Euclidean space, where an element t\in T can represent a point in space. But in general more results and theorems are possible for stochastic processes when the index set is ordered.

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There are other ways to consider a stochastic process, with the above definition being considered the traditional one. For example, a stochastic process can be interpreted or defined as a <math>S^T</math>-valued random variable, where <math>S^T</math> is the space of all the possible <math>S</math>-valued functions of <math>t\in T</math> that map from the set <math>T</math> into the space <math>S</math>. of the stochastic process. Often this set is some subset of the real line, such as the natural numbers or an interval, giving the set <math>T</math> the interpretation of time. such as the Cartesian plane <math>R^2</math> or <math>n</math>-dimensional Euclidean space, where an element <math>t\in T</math> can represent a point in space. But in general more results and theorems are possible for stochastic processes when the index set is ordered.

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~~还有其他的方法来考虑随机过程，上面的定义被认为是传统的定义。例如，随机过程可以被解释或定义为~~ s ^ t 值随机变量，其中 s ^ t 是 t 中所有可能的 s 值函数的空间，这些函数从集合 t ~~映射到随机过程空间 s。这个集合通常是实数直线的一些子集，比如自然数或者区间，给予集合~~ t ~~时间的解释。例如笛卡尔平面~~ r ^ 2或 n 维欧氏空间，其中 t ~~中的元素~~ t 可以表示空间中的一个点。但是一般来说，当指标集是有序的时候，对于随机过程可能有更多的结果和定理。

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还有其他的方法来考虑随机过程，上面的定义被认为是传统的定义。例如，随机过程可以被解释或定义为一个 < math > s ^ t </math > 值随机变量，其中 < math > s ^ t </math > 是 t </math > 中所有可能的 < math > s </math > 值函数的空间，这些函数从集合 < math > t </math > 映射到空间 < math > s </math > 。随机过程。这个集合通常是实数线的一些子集，比如自然数或者区间，赋予集合 < math > t </math > 时间的解释。比如笛卡尔平面 < math > r ^ 2 </math > 或 < math > n </math > 维欧氏空间，其中 t </math > 中的一个元素 < t 可以表示空间中的一个点。但是一般来说，当指标集是有序的时候，对于随机过程可能有更多的结果和定理。

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The mathematical space S of a stochastic process is called its state space. This mathematical space can be defined using integers, real lines, n-dimensional Euclidean spaces, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take.

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The mathematical space <math>S</math> of a stochastic process is called its state space. This mathematical space can be defined using integers, real lines, <math>n</math>-dimensional Euclidean spaces, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take.

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随机过程的数学空间 s 称为状态空间。这个数学空间可以用整数、实线、 n 维欧氏空间、复平面或更抽象的数学空间来定义。状态空间使用元素定义，这些元素反映了随机过程可以采用的不同值。

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随机过程的数学空间 s 称为状态空间。这个数学空间可以使用整数、实数线、维欧氏空间、复平面或更抽象的数学空间来定义。状态空间使用元素定义，这些元素反映了随机过程可以采用的不同值。

[[Almost surely]], a sample path of a Wiener process is continuous everywhere but [[nowhere differentiable function|nowhere differentiable]]. It can be considered as a continuous version of the simple random walk.<ref name="Applebaum2004page1337">{{cite journal|last1=Applebaum|first1=David|title=Lévy processes: From probability to finance and quantum groups|journal=Notices of the AMS|volume=51|issue=11|year=2004|page=1337}}</ref><ref name="MörtersPeres2010page1">{{cite book|author1=Peter Mörters|author2=Yuval Peres|title=Brownian Motion|url=https://books.google.com/books?id=e-TbA-dSrzYC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48657-6|pages=1, 3}}</ref> The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled,<ref name="KaratzasShreve2014page61">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=61}}</ref><ref name="Shreve2004page93">{{cite book|author=Steven E. Shreve|title=Stochastic Calculus for Finance II: Continuous-Time Models|url=https://books.google.com/books?id=O8kD1NwQBsQC|year=2004|publisher=Springer Science & Business Media|isbn=978-0-387-40101-0|page=93}}</ref> which is the subject of [[Donsker's theorem]] or invariance principle, also known as the functional central limit theorem.<ref name="Kallenberg2002page225and260">{{cite book|author=Olav Kallenberg|title=Foundations of Modern Probability|url=https://books.google.com/books?id=L6fhXh13OyMC|year=2002|publisher=Springer Science & Business Media|isbn=978-0-387-95313-7|pages=225, 260}}</ref><ref name="KaratzasShreve2014page70">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=70}}</ref><ref name="MörtersPeres2010page131">{{cite book|author1=Peter Mörters|author2=Yuval Peres|title=Brownian Motion|url=https://books.google.com/books?id=e-TbA-dSrzYC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48657-6|page=131}}</ref>

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The Wiener process is a member of some important families of stochastic processes, including Markov processes, Lévy processes and Gaussian processes.<ref name="RogersWilliams2000page1"/><ref name="Applebaum2004page1337"/> The process also has many applications and is the main stochastic process used in stochastic calculus.<ref name="Klebaner2005">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7}}</ref><ref name="KaratzasShreve2014page">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2}}</ref> It plays a central role in quantitative finance,<ref name="Applebaum2004page1341">{{cite journal|last1=Applebaum|first1=David|title=Lévy processes: From probability to finance and quantum groups|journal=Notices of the AMS|volume=51|issue=11|year=2004|page=1341}}</ref><ref name="KarlinTaylor2012page340">{{cite book|author1=Samuel Karlin|author2=Howard E. Taylor|title=A First Course in Stochastic Processes|url=https://books.google.com/books?id=dSDxjX9nmmMC|year=2012|publisher=Academic Press|isbn=978-0-08-057041-9|page=340}}</ref> where it is used, for example, in the Black–Scholes–Merton model.<ref name="Klebaner2005page124">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|page=124}}</ref> The process is also used in different fields, including the majority of natural sciences as well as some branches of social sciences, as a mathematical model for various random phenomena.<ref name="Steele2012page29"/><ref name="KaratzasShreve2014page47">{{cite book|author1=Ioannis Karatzas|author2=Steven Shreve|title=Brownian Motion and Stochastic Calculus|url=https://books.google.com/books?id=w0SgBQAAQBAJ&pg=PT5|year=1991|publisher=Springer|isbn=978-1-4612-0949-2|page=47}}</ref><ref name="Wiersema2008page2">{{cite book|author=Ubbo F. Wiersema|title=Brownian Motion Calculus|url=https://books.google.com/books?id=0h-n0WWuD9cC|year=2008|publisher=John Wiley & Sons|isbn=978-0-470-02171-2|page=2}}</ref>

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A sample function is a single outcome of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process. More precisely, if \{X(t,\omega):t\in T \} is a stochastic process, then for any point \omega\in\Omega, the mapping

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A sample function is a single outcome of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process. More precisely, if <math>\{X(t,\omega):t\in T \}</math> is a stochastic process, then for any point <math>\omega\in\Omega</math>, the mapping

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样本函数是随机过程的单一结果，所以它是由每个随机过程的随机变量的单一可能值构成的。更确切地说，如果 ~~t 中的~~{ x (~~t，ω~~) : t }~~是随机过程，那么对于 ω 中的任意点 ω，映射~~

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样本函数是随机过程的单一结果，所以它是由每个随机过程的随机变量的单一可能值构成的。更确切地说，如果 < math > { x (t，Omega) : t in t } </math > 是随机过程，那么对于任意点 < math > Omega </math > ，映射就是

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< 中心 > < 数学 >

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< 中心 > < 数学 >

===Poisson process===

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is called a sample function, a realization, or, particularly when T is interpreted as time, a sample path of the stochastic process \{X(t,\omega):t\in T \}. This means that for a fixed \omega\in\Omega, there exists a sample function that maps the index set T to the state space S. or path.

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is called a sample function, a realization, or, particularly when <math>T</math> is interpreted as time, a sample path of the stochastic process <math>\{X(t,\omega):t\in T \}</math>. This means that for a fixed <math>\omega\in\Omega</math>, there exists a sample function that maps the index set <math>T</math> to the state space <math>S</math>. or path.

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~~被称为样例函数，一个实现，或者，特别是当~~ t ~~被解释为时间时，随机过程~~{ x (t，omega) : t in t }~~的样例路径。这意味着，对于~~ Omega ~~中的一个固定 ω，存在一个示例函数，该函数将索引集~~ t 映射到状态空间 s ~~或路径。~~

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被称为一个样本函数，一个实现，或者，特别是当 < math > t </math > 被解释为时间，一个随机过程 < math > { x (t，omega) : t in t } </math > 的样本路径。这意味着对于 Omega </math > 中的一个固定的 < math > Omega，存在一个示例函数，它将索引集 < math > t </math > 映射到状态空间 < math > s </math > 。或者路径。

The Poisson process is a stochastic process that has different forms and definitions.<ref name="Tijms2003page1">{{cite book|author=Henk C. Tijms|title=A First Course in Stochastic Models|url=https://books.google.com/books?id=eBeNngEACAAJ|year=2003|publisher=Wiley|isbn=978-0-471-49881-0|pages=1, 2}}</ref><ref name="DaleyVere-Jones2006chap2">{{cite book|author1=D.J. Daley|author2=D. Vere-Jones|title=An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods|url=https://books.google.com/books?id=6Sv4BwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21564-8|pages=19–36}}</ref> It can be defined as a counting process, which is a stochastic process that represents the random number of points or events up to some time. The number of points of the process that are located in the interval from zero to some given time is a Poisson random variable that depends on that time and some parameter. This process has the natural numbers as its state space and the non-negative numbers as its index set. This process is also called the Poisson counting process, since it can be interpreted as an example of a counting process.<ref name="Tijms2003page1"/>

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If a Poisson process is defined with a single positive constant, then the process is called a homogeneous Poisson process.<ref name="Tijms2003page1"/><ref name="PinskyKarlin2011">{{cite book|author1=Mark A. Pinsky|author2=Samuel Karlin|title=An Introduction to Stochastic Modeling|url=https://books.google.com/books?id=PqUmjp7k1kEC|year=2011|publisher=Academic Press|isbn=978-0-12-381416-6|page=241}}</ref> The homogeneous Poisson process is a member of important classes of stochastic processes such as Markov processes and Lévy processes.<ref name="Applebaum2004page1337"/>

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An increment of a stochastic process is the difference between two random variables of the same stochastic process. For a stochastic process with an index set that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period. For example, if \{X(t):t\in T \} is a stochastic process with state space S and index set T=[0,\infty), then for any two non-negative numbers t_1\in [0,\infty) and t_2\in [0,\infty) such that t_1\leq t_2, the difference X_{t_2}-X_{t_1} is a S-valued random variable known as an increment.

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An increment of a stochastic process is the difference between two random variables of the same stochastic process. For a stochastic process with an index set that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period. For example, if <math>\{X(t):t\in T \}</math> is a stochastic process with state space <math>S</math> and index set <math>T=[0,\infty)</math>, then for any two non-negative numbers <math>t_1\in [0,\infty)</math> and <math>t_2\in [0,\infty)</math> such that <math>t_1\leq t_2</math>, the difference <math>X_{t_2}-X_{t_1}</math> is a <math>S</math>-valued random variable known as an increment.

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一个随机过程的增量是同一个随机过程的两个随机变量之间的差。对于一个索引集可以被解释为时间的随机过程，增量是随机过程在一定时间段内的变化量。例如，如果{ x (t) : t 在 t }~~中是一个状态空间~~ s ~~和索引集~~ t = [0，infty~~)的随机过程，那么对于任意两个非负数~~ t _ 1在[0，infty)和 t _ 2在[0，infty~~)中，使得~~ t _ 1 leq t _ ~~2，差~~ x _ { t _ 2}-x _ { t _ 1}是一个 s ~~值随机变量，称为增量。~~

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一个随机过程的增量是同一个随机过程的两个随机变量之间的差。对于一个索引集可以被解释为时间的随机过程，增量是随机过程在一定时间段内的变化量。例如，如果 < math > { x (t) : t in t } </math > 是一个状态空间 < math > s </math > 并且索引设置为 < math > t = [0，infty ] </math > ，那么对于[0，infty ] </math > 中的任意两个非负数 t _ 1和[0，infty ] </math > </math > t _ 2在[0，infty ] </math </math > 这样 < t _ 1 leq t _ 2 </math > ,差值 < math > x _ { t _ 2}-x _ { t _ 1} </math > 是一个 < math > s </math >-valued 随机变量，称为递增量。

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The homogeneous Poisson process can be defined and generalized in different ways. It can be defined such that its index set is the real line, and this stochastic process is also called the stationary Poisson process.<ref name="Kingman1992page38">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=38}}</ref><ref name="DaleyVere-Jones2006page19">{{cite book|author1=D.J. Daley|author2=D. Vere-Jones|title=An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods|url=https://books.google.com/books?id=6Sv4BwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21564-8|page=19}}</ref> If the parameter constant of the Poisson process is replaced with some non-negative integrable function of <math>t</math>, the resulting process is called an inhomogeneous or nonhomogeneous Poisson process, where the average density of points of the process is no longer constant.<ref name="Kingman1992page22">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=22}}</ref> Serving as a fundamental process in queueing theory, the Poisson process is an important process for mathematical models, where it finds applications for models of events randomly occurring in certain time windows.<ref name="KarlinTaylor2012page118">{{cite book|author1=Samuel Karlin|author2=Howard E. Taylor|title=A First Course in Stochastic Processes|url=https://books.google.com/books?id=dSDxjX9nmmMC|year=2012|publisher=Academic Press|isbn=978-0-08-057041-9|pages=118, 119}}</ref><ref name="Kleinrock1976page61">{{cite book|author=Leonard Kleinrock|title=Queueing Systems: Theory|url=https://archive.org/details/queueingsystems00klei|url-access=registration|year=1976|publisher=Wiley|isbn=978-0-471-49110-1|page=[https://archive.org/details/queueingsystems00klei/page/61 61]}}</ref>

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For a measurable subset B of S^T, the pre-image of X gives

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For a measurable subset <math>B</math> of <math>S^T</math>, the pre-image of <math>X</math> gives

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对于 s ^ t ~~的一个可测子集 b，x 的前象给出~~

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对于 < math > s ^ t </math > 的可测子集 < math > b </math > ，< math > x </math > 的前映像给出了

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< 中心 > < 数学 >

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< 中心 > < 数学 >

Defined on the real line, the Poisson process can be interpreted as a stochastic process,<ref name="Applebaum2004page1337"/><ref name="Rosenblatt1962page94">{{cite book|author=Murray Rosenblatt|title=Random Processes|url=https://archive.org/details/randomprocesses00rose_0|url-access=registration|year=1962|publisher=Oxford University Press|page=[https://archive.org/details/randomprocesses00rose_0/page/94 94]}}</ref> among other random objects.<ref name="Haenggi2013page10and18">{{cite book|author=Martin Haenggi|title=Stochastic Geometry for Wireless Networks|url=https://books.google.com/books?id=CLtDhblwWEgC|year=2013|publisher=Cambridge University Press|isbn=978-1-107-01469-5|pages=10, 18}}</ref><ref name="ChiuStoyan2013page41and108">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|pages=41, 108}}</ref> But then it can be defined on the <math>n</math>-dimensional Euclidean space or other mathematical spaces,<ref name="Kingman1992page11">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=11}}</ref> where it is often interpreted as a random set or a random counting measure, instead of a stochastic process.<ref name="Haenggi2013page10and18"/><ref name="ChiuStoyan2013page41and108"/> In this setting, the Poisson process, also called the Poisson point process, is one of the most important objects in probability theory, both for applications and theoretical reasons.<ref name="Stirzaker2000"/><ref name="Streit2010page1">{{cite book|author=Roy L. Streit|title=Poisson Point Processes: Imaging, Tracking, and Sensing|url=https://books.google.com/books?id=KAWmFYUJ5zsC&pg=PA11|year=2010|publisher=Springer Science & Business Media|isbn=978-1-4419-6923-1|page=1}}</ref> But it has been remarked that the Poisson process does not receive as much attention as it should, partly due to it often being considered just on the real line, and not on other mathematical spaces.<ref name="Streit2010page1"/><ref name="Kingman1992pagev">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|year=1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|page=v}}</ref>

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==Definitions==

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so the law of a X can be written as:

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so the law of a <math>X</math> can be written as:

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所以 x 的定律可以写成:

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所以 a 的定律可以写成:

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For a stochastic process X with law \mu, its finite-dimensional distributions are defined as:

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For a stochastic process <math>X</math> with law <math>\mu</math>, its finite-dimensional distributions are defined as:

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~~对于具有 μ 定律的随机过程 x，其有限维分布定义为~~:

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对于一个随机过程，其有限维分布被定义为:

In other words, for a given probability space <math>(\Omega, \mathcal{F}, P)</math> and a measurable space <math>(S,\Sigma)</math>, a stochastic process is a collection of <math>S</math>-valued random variables, which can be written as:<ref name="Florescu2014page293">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|page=293}}</ref>

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< 中心 > < 数学 >

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</math></center>

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where n\geq 1 is a counting number and each set t_i is a non-empty finite subset of the index set T, so each t_i\subset T, which means that t_1,\dots,t_n is any finite collection of subsets of the index set T.

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where <math>n\geq 1</math> is a counting number and each set <math>t_i</math> is a non-empty finite subset of the index set <math>T</math>, so each <math>t_i\subset T</math>, which means that <math>t_1,\dots,t_n</math> is any finite collection of subsets of the index set <math>T</math>.

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其中 n ~~geq1是一个计数数，每个集合 ti~~ 是指数集 t ~~的一个非空有限子集，因此每个~~ t i 子集 ~~t，意味着~~ t _ 1，点，t _ n 是指数集 t ~~的任意有限子集。~~

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其中 < math > n geq 1 </math > 是一个计数数字，每个集 < math > t i </math > 是指数集 < math > t </math > 的非空有限子集，因此每个 < math > t i 子集 t </math > ，这意味着 < math > t _ 1，点，t _ n </math > 是指数集 < math > t </math > 的任何有限子集。

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Historically, in many problems from the natural sciences a point <math>t\in T</math> had the meaning of time, so <math>X(t)</math> is a random variable representing a value observed at time <math>t</math>.<ref name="Borovkov2013page528">{{cite book|author=Alexander A. Borovkov|authorlink=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=528}}</ref> A stochastic process can also be written as <math> \{X(t,\omega):t\in T \}</math> to reflect that it is actually a function of two variables, <math>t\in T</math> and <math>\omega\in \Omega</math>.<ref name="Lamperti1977page1"/><ref name="LindgrenRootzen2013page11">{{cite book|author1=Georg Lindgren|author2=Holger Rootzen|author3=Maria Sandsten|title=Stationary Stochastic Processes for Scientists and Engineers|url=https://books.google.com/books?id=FYJFAQAAQBAJ&pg=PR1|year=2013|publisher=CRC Press|isbn=978-1-4665-8618-5|pages=11}}</ref>

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For any measurable subset C of the n-fold Cartesian power S^n=S\times\dots \times S, the finite-dimensional distributions of a stochastic process X can be written as: But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.

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For any measurable subset <math>C</math> of the <math>n</math>-fold Cartesian power <math>S^n=S\times\dots \times S</math>, the finite-dimensional distributions of a stochastic process <math>X</math> can be written as: But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.

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对于 n ~~次笛卡尔幂~~ s ^ n = s ~~乘以点~~ s ~~的任意可测子集 c，随机过程~~ x 的有限维分布可以写成: ~~但是平稳性的概念也存在于点过程和随机场，其中指数集不解释为时间。~~

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对于任何可测量的子集 < math > c </math > n </math >-fold 笛卡尔幂 < math > s ^ n = s 乘以 s </math > ，一个随机过程 < math > x </math > 的有限维分布可以写成: 但是平稳性的概念也存在于点过程和随机场，其中指数集不被解释为时间。

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There are other ways to consider a stochastic process, with the above definition being considered the traditional one.<ref name="RogersWilliams2000page121">{{cite book|author1=L. C. G. Rogers|author2=David Williams|title=Diffusions, Markov Processes, and Martingales: Volume 1, Foundations|url=https://books.google.com/books?id=W0ydAgAAQBAJ&pg=PA1|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|pages=121, 122}}</ref><ref name="Asmussen2003page408">{{cite book|author=Søren Asmussen|title=Applied Probability and Queues|url=https://books.google.com/books?id=BeYaTxesKy0C|year=2003|publisher=Springer Science & Business Media|isbn=978-0-387-00211-8|page=408}}</ref> For example, a stochastic process can be interpreted or defined as a <math>S^T</math>-valued random variable, where <math>S^T</math> is the space of all the possible <math>S</math>-valued [[Function (mathematics)|functions]] of <math>t\in T</math> that [[Map (mathematics)|map]] from the set <math>T</math> into the space <math>S</math>.<ref name="Kallenberg2002page24"/><ref name="RogersWilliams2000page121"/>

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When the index set T can be interpreted as time, a stochastic process is said to be stationary if its finite-dimensional distributions are invariant under translations of time. This type of stochastic process can be used to describe a physical system that is in steady state, but still experiences random fluctuations. A sequence of random variables forms a stationary stochastic process only if the random variables are identically distributed. Khinchin introduced the related concept of stationarity in the wide sense, which has other names including covariance stationarity or stationarity in the broad sense.

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When the index set <math>T</math> can be interpreted as time, a stochastic process is said to be stationary if its finite-dimensional distributions are invariant under translations of time. This type of stochastic process can be used to describe a physical system that is in steady state, but still experiences random fluctuations. A sequence of random variables forms a stationary stochastic process only if the random variables are identically distributed. Khinchin introduced the related concept of stationarity in the wide sense, which has other names including covariance stationarity or stationarity in the broad sense.

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当指数集 t 可以解释为时间时，如果一个随机过程的有限维分布在时间平移下是不变的，则称其为稳定的。这种类型的随机过程可以用来描述一个处于稳定状态但仍然经历随机波动的物理系统。只有当随机变量是同分布的时候，一系列随机变量才会形成一个平稳的随机过程。Khinchin 提出了广义平稳性的相关概念，广义的协方差平稳性或平稳性又有其他名称。

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当指数集 < math > t </math > 可以被解释为时间时，如果一个随机过程的有限维分布在时间平移下是不变的，那么它就是静止的。这种类型的随机过程可以用来描述一个处于稳定状态但仍然经历随机波动的物理系统。只有当随机变量是同分布的时候，一系列随机变量才会形成一个平稳的随机过程。Khinchin 提出了广义平稳性的相关概念，广义的协方差平稳性或平稳性又有其他名称。

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The set <math>T</math> is called the '''index set'''<ref name="Parzen1999"/><ref name="Florescu2014page294"/> or '''parameter set'''<ref name="Lamperti1977page1"/><ref name="Skorokhod2005page93">{{cite book|author=Valeriy Skorokhod|title=Basic Principles and Applications of Probability Theory|url=https://books.google.com/books?id=dQkYMjRK3fYC|year=2005|publisher=Springer Science & Business Media|isbn=978-3-540-26312-8|pages=93, 94}}</ref> of the stochastic process. Often this set is some subset of the [[real line]], such as the [[natural numbers]] or an interval, giving the set <math>T</math> the interpretation of time.<ref name="doob1953stochasticP46to47"/> In addition to these sets, the index set <math>T</math> can be other linearly ordered sets or more general mathematical sets,<ref name="doob1953stochasticP46to47"/><ref name="Billingsley2008page482">{{cite book|author=Patrick Billingsley|title=Probability and Measure|url=https://books.google.com/books?id=QyXqOXyxEeIC|year=2008|publisher=Wiley India Pvt. Limited|isbn=978-81-265-1771-8|page=482}}</ref> such as the Cartesian plane <math>R^2</math> or <math>n</math>-dimensional Euclidean space, where an element <math>t\in T</math> can represent a point in space.<ref name="KarlinTaylor2012page27">{{cite book|author1=Samuel Karlin|author2=Howard E. Taylor|title=A First Course in Stochastic Processes|url=https://books.google.com/books?id=dSDxjX9nmmMC|year=2012|publisher=Academic Press|isbn=978-0-08-057041-9|page=27}}</ref><ref>{{cite book|author1=Donald L. Snyder|author2=Michael I. Miller|title=Random Point Processes in Time and Space|url=https://books.google.com/books?id=c_3UBwAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4612-3166-0|page=25}}</ref> But in general more results and theorems are possible for stochastic processes when the index set is ordered.<ref name="Skorokhod2005page104">{{cite book|author=Valeriy Skorokhod|title=Basic Principles and Applications of Probability Theory|url=https://books.google.com/books?id=dQkYMjRK3fYC|year=2005|publisher=Springer Science & Business Media|isbn=978-3-540-26312-8|page=104}}</ref>

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A filtration is an increasing sequence of sigma-algebras defined in relation to some probability space and an index set that has some total order relation, such as in the case of the index set being some subset of the real numbers. More formally, if a stochastic process has an index set with a total order, then a filtration \{\mathcal{F}_t\}_{t\in T} , on a probability space (\Omega, \mathcal{F}, P) is a family of sigma-algebras such that \mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F} for all s \leq t, where t, s\in T and \leq denotes the total order of the index set T.

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A filtration is an increasing sequence of sigma-algebras defined in relation to some probability space and an index set that has some total order relation, such as in the case of the index set being some subset of the real numbers. More formally, if a stochastic process has an index set with a total order, then a filtration <math>\{\mathcal{F}_t\}_{t\in T} </math>, on a probability space <math>(\Omega, \mathcal{F}, P)</math> is a family of sigma-algebras such that <math> \mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F} </math> for all <math>s \leq t</math>, where <math>t, s\in T</math> and <math>\leq</math> denotes the total order of the index set <math>T</math>.

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过滤是一个增加序列的 sigma-代数定义关于一些概率空间和一个索引集，有一些总序关系，例如在情况下的索引集是一些子集的实数。更正式地说，如果一个随机过程有一个总序的索引集，那么在一个概率空间(Omega，mathcal { f } ，p)上的一个过滤{ mathcal { f } _ t } _ { t in t }是一个 σ 代数族，使得对于所有的 s leq t，数学{ f } _ t 子序列 q { f } ，其中 t，s in t 和 leq 表示索引集 t 的总序列。

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过滤是一个增加序列的 sigma-代数定义关于一些概率空间和一个索引集，有一些总序关系，例如在情况下的索引集是一些子集的实数。更正式的说法是，如果一个随机过程有一个总序的索引集，那么在一个总序的索引集上，对一个概率空间的索引集进行一次过滤，这样的索引集就是一个总序的索引集，这样的索引集的总序就是数学的。

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The [[mathematical space]] <math>S</math> of a stochastic process is called its ''state space''. This mathematical space can be defined using [[integer]]s, [[real line]]s, <math>n</math>-dimensional [[Euclidean space]]s, complex planes, or more abstract mathematical spaces. The state space is defined using elements that reflect the different values that the stochastic process can take.<ref name="doob1953stochasticP46to47"/><ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/><ref name="Florescu2014page294">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|pages=294, 295}}</ref><ref name="Brémaud2014page120">{{cite book|author=Pierre Brémaud|title=Fourier Analysis and Stochastic Processes|url=https://books.google.com/books?id=dP2JBAAAQBAJ&pg=PA1|year=2014|publisher=Springer|isbn=978-3-319-09590-5|page=120}}</ref>

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A modification of a stochastic process is another stochastic process, which is closely related to the original stochastic process. More precisely, a stochastic process X that has the same index set T, set space S, and probability space (\Omega,{\cal F},P) as another stochastic process Y is said to be a modification of Y if for all t\in T the following

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A modification of a stochastic process is another stochastic process, which is closely related to the original stochastic process. More precisely, a stochastic process <math>X</math> that has the same index set <math>T</math>, set space <math>S</math>, and probability space <math>(\Omega,{\cal F},P)</math> as another stochastic process <math>Y</math> is said to be a modification of <math>Y</math> if for all <math>t\in T</math> the following

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随机过程的一个修改是另一个随机过程，这是密切相关的原始随机过程。更确切地说，如果一个随机过程 x 与另一个随机过程 y 的索引集 t、索引空间 s 和索引概率空间(Omega，{ cal f } ，p)相同，那么这个 x 就是 y 的修改，如果 t 中的所有 t 都是 y 的话

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随机过程的一个修改是另一个随机过程，这是密切相关的原始随机过程。更确切地说，一个随机过程，一个同样的指数集，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个空间，一个

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< 中心 > < 数学 >

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< 中心 > < 数学 >

===Sample function===

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</math></center>

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Instead of modification, the term version is also used, however some authors use the term version when two stochastic processes have the same finite-dimensional distributions, but they may be defined on different probability spaces, so two processes that are modifications of each other, are also versions of each other, in the latter sense, but not the converse. The theorem can also be generalized to random fields so the index set is n-dimensional Euclidean space as well as to stochastic processes with metric spaces as their state spaces.

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Instead of modification, the term version is also used, however some authors use the term version when two stochastic processes have the same finite-dimensional distributions, but they may be defined on different probability spaces, so two processes that are modifications of each other, are also versions of each other, in the latter sense, but not the converse. The theorem can also be generalized to random fields so the index set is <math>n</math>-dimensional Euclidean space as well as to stochastic processes with metric spaces as their state spaces.

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不同的概率空间可以定义不同的两个随机过程，因此两个相互修正的过程，在后一种意义上也是相互修正的过程，但不是相反。这个定理也可以推广到随机场，因此指标集是 n ~~维欧氏空间，也可以推广到以度量空间为状态空间的随机过程。~~

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不同的概率空间可以定义不同的两个随机过程，因此两个相互修正的过程，在后一种意义上也是相互修正的过程，但不是相反。这个定理也可以推广到随机场，因此指数集是 < math > n </math > 维欧氏空间，以及以度量空间为状态空间的随机过程。

is called a sample function, a '''realization''', or, particularly when <math>T</math> is interpreted as time, a '''sample path''' of the stochastic process <math>\{X(t,\omega):t\in T \}</math>.<ref name="RogersWilliams2000page121b">{{cite book|author1=L. C. G. Rogers|author2=David Williams|title=Diffusions, Markov Processes, and Martingales: Volume 1, Foundations|url=https://books.google.com/books?id=W0ydAgAAQBAJ&pg=PA1|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|pages=121–124}}</ref> This means that for a fixed <math>\omega\in\Omega</math>, there exists a sample function that maps the index set <math>T</math> to the state space <math>S</math>.<ref name="Lamperti1977page1"/> Other names for a sample function of a stochastic process include '''trajectory''', '''path function'''<ref name="Billingsley2008page493">{{cite book|author=Patrick Billingsley|title=Probability and Measure|url=https://books.google.com/books?id=QyXqOXyxEeIC|year=2008|publisher=Wiley India Pvt. Limited|isbn=978-81-265-1771-8|page=493}}</ref> or '''path'''.<ref name="Øksendal2003page10">{{cite book|author=Bernt Øksendal|title=Stochastic Differential Equations: An Introduction with Applications|url=https://books.google.com/books?id=VgQDWyihxKYC|year=2003|publisher=Springer Science & Business Media|isbn=978-3-540-04758-2|page=10}}</ref>

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===Increment===

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Two stochastic processes X and Y defined on the same probability space (\Omega,\mathcal{F},P) with the same index set T and set space S are said be indistinguishable if the following

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Two stochastic processes <math>X</math> and <math>Y</math> defined on the same probability space <math>(\Omega,\mathcal{F},P)</math> with the same index set <math>T</math> and set space <math>S</math> are said be indistinguishable if the following

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~~在同一概率空间上定义的具有相同指数集~~ t ~~和集空间~~ s ~~的随机过程 x 和 y，如果下列情况，则无法区分~~

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两个随机过程 < math > x </math > 和 < math > y </math > 定义在同一个概率空间 < math > (Omega，cal { f } ，p) </math > 具有相同的指数集 < math > t </math > 和集合空间 < math > s </math > 如果下列情况，这两个随机过程是无法区分的

An '''increment''' of a stochastic process is the difference between two random variables of the same stochastic process. For a stochastic process with an index set that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period. For example, if <math>\{X(t):t\in T \}</math> is a stochastic process with state space <math>S</math> and index set <math>T=[0,\infty)</math>, then for any two non-negative numbers <math>t_1\in [0,\infty)</math> and <math>t_2\in [0,\infty)</math> such that <math>t_1\leq t_2</math>, the difference <math>X_{t_2}-X_{t_1}</math> is a <math>S</math>-valued random variable known as an increment.<ref name="KarlinTaylor2012page27"/><ref name="Applebaum2004page1337"/> When interested in the increments, often the state space <math>S</math> is the real line or the natural numbers, but it can be <math>n</math>-dimensional Euclidean space or more abstract spaces such as [[Banach space]]s.<ref name="Applebaum2004page1337"/>

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< 中心 > < 数学 >

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< 中心 > < 数学 >

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</math></center>

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More precisely, a real-valued continuous-time stochastic process X with a probability space (\Omega,{\cal F},P) is separable if its index set T has a dense countable subset U\subset T and there is a set \Omega_0 \subset \Omega of probability zero, so P(\Omega_0)=0, such that for every open set G\subset T and every closed set F\subset \textstyle R =(-\infty,\infty) , the two events \{ X_t \in F \text{ for all } t \in G\cap U\} and \{ X_t \in F \text{ for all } t \in G\} differ from each other at most on a subset of \Omega_0.

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More precisely, a real-valued continuous-time stochastic process <math>X</math> with a probability space <math>(\Omega,{\cal F},P)</math> is separable if its index set <math>T</math> has a dense countable subset <math>U\subset T</math> and there is a set <math>\Omega_0 \subset \Omega</math> of probability zero, so <math>P(\Omega_0)=0</math>, such that for every open set <math>G\subset T</math> and every closed set <math>F\subset \textstyle R =(-\infty,\infty) </math>, the two events <math>\{ X_t \in F \text{ for all } t \in G\cap U\}</math> and <math>\{ X_t \in F \text{ for all } t \in G\}</math> differ from each other at most on a subset of <math>\Omega_0</math>.

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~~更精确地说，如果指数集 t 有一个稠密可数子集 u 子集 t 且存在一个概率为零的随机过程 ω，则实值连续时刻具有概率空间~~(ω，{ cal f } ，p)~~的 x 是可分的~~,所以 p (Omega _ 0) = ~~0，对于每个开集~~ g 子集 t 和每个闭集 f ~~子集 textstyle~~ r = (- infty，infty) ~~，两个事件~~{ x _ t 在 f text { for all } t 在 g 开头 u }和{ x _ t 在 f text { for all } t 在 ~~Omega _ 0的子集上最多不同。~~

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更确切地说，一个带有随机过程的实值连续时间子集 x < math > (Omega，{ cal f } ，p) </math > 是可分的，如果它的指数集 < math > t </math > 有一个稠密的可数子集 < math > u t </math > 并且存在一个集合 < math > Omega 0子集 ω </math > 概率为0,所以 < math > p (Omega _ 0) = 0 </math > ，对于每个开集 < math > g 子集 t </math > 和每个闭集 < math > f 子集文本样式 r = (- infty，infty) </math > ,两个事件 < math > > { x _ t in f text { for all } t in g cap u } </math > 和 < math > { x _ t in f text { for all } t in g } </math > 在 < math > 的子集上最多不同。

where <math>P</math> is a probability measure, the symbol <math>\circ </math> denotes function composition and <math>X^{-1}</math> is the pre-image of the measurable function or, equivalently, the <math>S^T</math>-valued random variable <math>X</math>, where <math>S^T</math> is the space of all the possible <math>S</math>-valued functions of <math>t\in T</math>, so the law of a stochastic process is a probability measure.<ref name="Kallenberg2002page24"/><ref name="RogersWilliams2000page121"/><ref name="FrizVictoir2010page571"/><ref name="Resnick2013page40">{{cite book|author=Sidney I. Resnick|title=Adventures in Stochastic Processes|url=https://books.google.com/books?id=VQrpBwAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4612-0387-2|pages=40–41}}</ref>

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The definition of separability can also be stated for other index sets and state spaces, such as in the case of random fields, where the index set as well as the state space can be n-dimensional Euclidean space. A theorem by Doob, sometimes known as Doob's separability theorem, says that any real-valued continuous-time stochastic process has a separable modification. Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line.

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The definition of separability can also be stated for other index sets and state spaces, such as in the case of random fields, where the index set as well as the state space can be <math>n</math>-dimensional Euclidean space. A theorem by Doob, sometimes known as Doob's separability theorem, says that any real-valued continuous-time stochastic process has a separable modification. Versions of this theorem also exist for more general stochastic processes with index sets and state spaces other than the real line.

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~~可分性的定义也适用于其他的索引集和状态空间，例如在随机场的情况下，索引集和状态空间都可以是~~ n ~~维欧氏空间。Doob~~ 的一个定理，有时也被称为 Doob 的可分性定理，说任何实值连续时间随机过程都有一个可分的修正。这个定理的版本也存在于更一般的索引集和状态空间的随机过程，而不是实线。

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可分性的定义也适用于其他索引集和状态空间，例如在随机场的情况下，索引集和状态空间可以是 < math > n </math >-dimensional Euclidean 空间。Doob 的一个定理，有时也被称为 Doob 的可分性定理，说任何实值连续时间随机过程都有一个可分的修正。这个定理的版本也存在于更一般的索引集和状态空间的随机过程，而不是实线。

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Two stochastic processes \left\{X_t\right\} and \left\{Y_t\right\} are called uncorrelated if their cross-covariance \operatorname{K}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = \operatorname{E} \left[ \left( X(t_1)- \mu_X(t_1) \right) \left( Y(t_2)- \mu_Y(t_2) \right) \right] is zero for all times. Formally:

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Two stochastic processes <math>\left\{X_t\right\}</math> and <math>\left\{Y_t\right\}</math> are called uncorrelated if their cross-covariance <math>\operatorname{K}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = \operatorname{E} \left[ \left( X(t_1)- \mu_X(t_1) \right) \left( Y(t_2)- \mu_Y(t_2) \right) \right]</math> is zero for all times. Formally:

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~~如果左~~{ x _ t 右}和左{ y _ t 右}~~的互协方差操作数名~~{ k }{ mathbf { x } ~~mathbf~~ { y }(t _ 1，t _ 2) = ~~操作数名~~{ e }左[ x (t _ 1)-mu _ x (t _ 1)右]左(y (t _ 2)-~~mu _~~ y (t _ 2)右]~~始终为零，则称左{ x _ t 右}和左{ y _ t 右}是不相关的。形式上~~:

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两个随机过程 < math > 左{ x _ t 右} </math > 和 < math > 左{ y _ t 右} </math > 如果它们的互协方差 < math > 操作者名{ k }{ mathbf { x }{ y }}(t _ 1，t _ 2) = 操作者名{ e }左(x (t _ 1)-mu _ x (t _ 1)右)左(y (t _ 2)-y _ 2)右] </math > 始终为零，则称为不相关过程。形式上:

X^{-1}(B)=\{\omega\in \Omega: X(\omega)\in B \},

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</math></center>

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\left\{X_t\right\},\left\{Y_t\right\} \text{ uncorrelated} \quad \iff \quad \operatorname{K}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = 0 \quad \forall t_1,t_2.

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<math>\left\{X_t\right\},\left\{Y_t\right\} \text{ uncorrelated} \quad \iff \quad \operatorname{K}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = 0 \quad \forall t_1,t_2</math>.

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左{ x _ t 右} ，左{ y _ t 右}文本{ uncorrelated }~~四边形函数名~~{ k }{ mathbf { x } ~~mathbf~~ { y }(t _ 1，t _ 2) = ~~0对所有~~ t _ 1，t _ 2。

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左{ x _ t 右} ，左{ y _ t 右}文本{ uncorrelated }四匹配四匹配操作器名{ k }{ mathbf { x }{ y }(t _ 1，t _ 2) = 0四匹配所有 t _ 1，t _ 2 </math > 。

so the law of a <math>X</math> can be written as:<ref name="Lamperti1977page1"/>

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\mu(B)=P(\{\omega\in \Omega: X(\omega)\in B \}).

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If two stochastic processes X and Y are independent, then they are also uncorrelated. Such functions are known as càdlàg or cadlag functions, based on the acronym of the French expression continue à droite, limite à gauche, due to the functions being right-continuous with left limits. A Skorokhod function space, introduced by Anatoliy Skorokhod, The notation of this function space can also include the interval on which all the càdlàg functions are defined, so, for example, D[0,1] denotes the space of càdlàg functions defined on the unit interval [0,1].

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If two stochastic processes <math>X</math> and <math>Y</math> are independent, then they are also uncorrelated. Such functions are known as càdlàg or cadlag functions, based on the acronym of the French expression continue à droite, limite à gauche, due to the functions being right-continuous with left limits. A Skorokhod function space, introduced by Anatoliy Skorokhod, The notation of this function space can also include the interval on which all the càdlàg functions are defined, so, for example, <math>D[0,1]</math> denotes the space of càdlàg functions defined on the unit interval <math>[0,1]</math>.

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如果两个随机过程 x 和 y 是独立的，那么它们也是不相关的。这种函数称为 càdlàg 或 cadlag 函数，由法语表达式 continue à droite，limite à gauche 的首字母缩写而来，因为这些函数是右连续的，有左限制。由 Anatoliy Skorokhod 引入的 Skorokhod ~~函数空间，这个函数空间的符号也可以包括定义所有 càdlàg 函数的区间，因此，例如，d~~ [0,1]表示在单位区间[0,1]~~上定义的 càdlà g 函数的空间。~~

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如果两个随机过程 < math > x </math > 和 < math > y </math > 是独立的，那么它们也是不相关的。这种函数称为 càdlàg 或 cadlag 函数，由法语表达式 continue à droite，limite à gauche 的首字母缩写而来，因为这些函数是右连续的，有左限制。由 Anatoliy Skorokhod 引入的 Skorokhod 函数空间，这个函数空间的符号也可以包括定义所有函数的区间，因此，例如，< math > d [0,1] </math > 表示在单位区间 < math > [0,1] </math > 上定义的函数的空间。

</math></center>

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====Stationarity====

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The concept of the Markov property was originally for stochastic processes in continuous and discrete time, but the property has been adapted for other index sets such as n-dimensional Euclidean space, which results in collections of random variables known as Markov random fields.

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The concept of the Markov property was originally for stochastic processes in continuous and discrete time, but the property has been adapted for other index sets such as <math>n</math>-dimensional Euclidean space, which results in collections of random variables known as Markov random fields.

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~~马尔可夫性的概念最初是针对连续时间和离散时间的随机过程，但这一性质已经适用于其他指标集，如~~ n 维欧氏空间，这导致了被称为马尔可夫随机场的随机变量集合。

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马尔可夫性的概念最初是用于连续和离散时间的随机过程，但是这个性质已经适用于其他指标集，如 < math > n </math > 维欧氏空间，这导致了被称为马尔可夫随机场的随机变量集合。

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====Filtration====

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Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference. They have found applications in areas in probability theory such as queueing theory and Palm calculus and other fields such as economics and finance. These processes have many applications in fields such as finance, fluid mechanics, physics and biology. The main defining characteristics of these processes are their stationarity and independence properties, so they were known as processes with stationary and independent increments. In other words, a stochastic process X is a Lévy process if for n non-negatives numbers, 0\leq t_1\leq \dots \leq t_n, the corresponding n-1 increments

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Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference. They have found applications in areas in probability theory such as queueing theory and Palm calculus and other fields such as economics and finance. These processes have many applications in fields such as finance, fluid mechanics, physics and biology. The main defining characteristics of these processes are their stationarity and independence properties, so they were known as processes with stationary and independent increments. In other words, a stochastic process <math>X</math> is a Lévy process if for <math>n</math> non-negatives numbers, <math>0\leq t_1\leq \dots \leq t_n</math>, the corresponding <math>n-1</math> increments

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鞅在统计学中有许多应用，但有人指出，鞅的使用和应用并不象在统计学领域，特别是推论统计学统计学领域那样广泛。他们已经在排队论和 Palm 演算以及其他领域如经济和金融等概率论领域找到了应用。这些过程在金融、流体力学、物理学和生物学等领域有许多应用。这些过程的主要定义特征是它们的平稳性和独立性，因此它们被称为具有平稳增量和独立增量的过程。换句话说，一个随机过程 x 是一个 Lévy 过程，如果对 n ~~个非负数，0~~ leq t _ 1 leq dots leq t ~~_ n，相应的~~ n-~~1增量~~

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鞅在统计学中有许多应用，但有人指出，鞅的使用和应用并不象在统计学领域，特别是推论统计学统计学领域那样广泛。他们已经在排队论和 Palm 演算以及其他领域如经济和金融等概率论领域找到了应用。这些过程在金融、流体力学、物理学和生物学等领域有许多应用。这些过程的主要定义特征是它们的平稳性和独立性，因此它们被称为具有平稳增量和独立增量的过程。换句话说，如果对于 < math > n </math > 非负数，< math > 0 leq t 1 leq dots leq t n </math > ，相应的 < math > n-1 </math > 递增值是一个 Lévy 过程

A [[Filtration (probability theory)|filtration]] is an increasing sequence of sigma-algebras defined in relation to some probability space and an index set that has some [[total order]] relation, such as in the case of the index set being some subset of the real numbers. More formally, if a stochastic process has an index set with a total order, then a filtration <math>\{\mathcal{F}_t\}_{t\in T} </math>, on a probability space <math>(\Omega, \mathcal{F}, P)</math> is a family of sigma-algebras such that <math> \mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F} </math> for all <math>s \leq t</math>, where <math>t, s\in T</math> and <math>\leq</math> denotes the total order of the index set <math>T</math>.<ref name="Florescu2014page294"/> With the concept of a filtration, it is possible to study the amount of information contained in a stochastic process <math>X_t</math> at <math>t\in T</math>, which can be interpreted as time <math>t</math>.<ref name="Florescu2014page294"/><ref name="Williams1991page93"/> The intuition behind a filtration <math>\mathcal{F}_t</math> is that as time <math>t</math> passes, more and more information on <math>X_t</math> is known or available, which is captured in <math>\mathcal{F}_t</math>, resulting in finer and finer partitions of <math>\Omega</math>.<ref name="Klebaner2005page22">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|pages=22–23}}</ref><ref name="MörtersPeres2010page37">{{cite book|author1=Peter Mörters|author2=Yuval Peres|title=Brownian Motion|url=https://books.google.com/books?id=e-TbA-dSrzYC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48657-6|page=37}}</ref>

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< 中心 > < 数学 >

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< 中心 > < 数学 >

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holds. Two stochastic processes that are modifications of each other have the same finite-dimensional law<ref name="RogersWilliams2000page130">{{cite book|author1=L. C. G. Rogers|author2=David Williams|title=Diffusions, Markov Processes, and Martingales: Volume 1, Foundations|url=https://books.google.com/books?id=W0ydAgAAQBAJ&pg=PA356|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|page=130}}</ref> and they are said to be '''stochastically equivalent''' or '''equivalent'''.<ref name="Borovkov2013page530">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=530}}</ref>

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A point process is a collection of points randomly located on some mathematical space such as the real line, n-dimensional Euclidean space, or more abstract spaces. Sometimes the term point process is not preferred, as historically the word process denoted an evolution of some system in time, so a point process is also called a random point field. There are different interpretations of a point process, such a random counting measure or a random set. Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic process, though it has been remarked that the difference between point processes and stochastic processes is not clear. which corresponds to the index set in stochastic process terminology.}} on which it is defined, such as the real line or n-dimensional Euclidean space. Other stochastic processes such as renewal and counting processes are studied in the theory of point processes.

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A point process is a collection of points randomly located on some mathematical space such as the real line, <math>n</math>-dimensional Euclidean space, or more abstract spaces. Sometimes the term point process is not preferred, as historically the word process denoted an evolution of some system in time, so a point process is also called a random point field. There are different interpretations of a point process, such a random counting measure or a random set. Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic process, though it has been remarked that the difference between point processes and stochastic processes is not clear. which corresponds to the index set in stochastic process terminology.}} on which it is defined, such as the real line or <math>n</math>-dimensional Euclidean space. Other stochastic processes such as renewal and counting processes are studied in the theory of point processes.

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~~点过程是在一些数学空间(如实直线、~~ n 维欧氏空间或更多的抽象空间)上随机定位的点的集合。有时，词汇点过程并不是首选，因为历史上词汇过程表示某个系统在时间上的演变，所以点过程也称为随机点场。一个点过程有不同的解释，比如随机计数测度或随机集合。有些作者把点过程和随机过程过程看作是两个不同的对象，例如，点过程是一个随机的对象，它起源于或与随机过程过程相关联，尽管有人指出点过程和随机过程之间的区别并不清楚。它对应于随机过程术语中的索引集。}~~实线或~~ n ~~维欧几里德空间。在点过程理论中研究了更新和计数过程等其他随机过程。~~

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点过程是一个点的集合，这些点随机地分布在一些数学空间上，比如实数直线、 n 维欧氏空间或者更多的抽象空间。有时，词汇点过程并不是首选，因为历史上词汇过程表示某个系统在时间上的演变，所以点过程也称为随机点场。一个点过程有不同的解释，比如随机计数测度或随机集合。有些作者把点过程和随机过程过程看作是两个不同的对象，例如，点过程是一个随机的对象，它起源于或与随机过程过程相关联，尽管有人指出点过程和随机过程之间的区别并不清楚。它对应于随机过程术语中的索引集。}它被定义在其上，例如实线或者 < math > n </math > 维欧氏空间。在点过程理论中研究了更新和计数过程等其他随机过程。

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Two stochastic processes <math>\left\{X_t\right\}</math> and <math>\left\{Y_t\right\}</math> are called '''orthogonal''' if their cross-correlation <math>\operatorname{R}_{\mathbf{X}\mathbf{Y}}(t_1,t_2) = \operatorname{E}[X(t_1) \overline{Y(t_2)}]</math> is zero for all times.<ref name=KunIlPark/>{{rp|p. 142}} Formally:

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For random walks in n-dimensional integer lattices, George Pólya published in 1919 and 1921 work, where he studied the probability of a symmetric random walk returning to a previous position in the lattice. Pólya showed that a symmetric random walk, which has an equal probability to advance in any direction in the lattice, will return to a previous position in the lattice an infinite number of times with probability one in one and two dimensions, but with probability zero in three or higher dimensions.

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For random walks in <math>n</math>-dimensional integer lattices, George Pólya published in 1919 and 1921 work, where he studied the probability of a symmetric random walk returning to a previous position in the lattice. Pólya showed that a symmetric random walk, which has an equal probability to advance in any direction in the lattice, will return to a previous position in the lattice an infinite number of times with probability one in one and two dimensions, but with probability zero in three or higher dimensions.

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对于 n 维整数格中的随机游动，George Pólya ~~在1919年和1921年发表的著作中，研究了对称随机游动回到格中先前位置的概率。Pólya~~ 证明了对称随机游动，它在格子中向任何方向前进的概率相等，将无限次地回到格子中的一个先前的位置，概率为1在一维和2维，但概率为0在三维或更高维。

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对于 < math > n </math > 维整数格中的随机游动，George Pólya 在1919年和1921年发表的工作中，他研究了对称随机游动回到格中以前位置的概率。Pólya 证明了对称随机游动，它在格子中向任何方向前进的概率相等，将无限次地回到格子中的一个先前的位置，概率为1在一维和2维，但概率为0在三维或更高维。

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Although less used, the separability assumption is considered more general because every stochastic process has a separable version. For example, separability is assumed when constructing and studying random fields, where the collection of random variables is now indexed by sets other than the real line such as n-dimensional Euclidean space.

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Although less used, the separability assumption is considered more general because every stochastic process has a separable version. For example, separability is assumed when constructing and studying random fields, where the collection of random variables is now indexed by sets other than the real line such as <math>n</math>-dimensional Euclidean space.

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尽管很少使用，但是可分性假设被认为是更一般的，因为每个随机过程都有一个可分离的版本。例如，在构造和研究随机场时假设可分性，其中随机变量的集合现在由实线以外的集合索引，如 n 维欧氏空间。

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尽管很少使用，但是可分性假设被认为是更一般的，因为每个随机过程都有一个可分离的版本。例如，在构造和研究随机场时假设可分性，其中随机变量的集合现在由实线以外的集合索引，如 < math > n </math > 维欧氏空间。

Markov processes are stochastic processes, traditionally in [[Discrete time and continuous time|discrete or continuous time]], that have the Markov property, which means the next value of the Markov process depends on the current value, but it is conditionally independent of the previous values of the stochastic process. In other words, the behavior of the process in the future is stochastically independent of its behavior in the past, given the current state of the process.<ref name="Serfozo2009page2">{{cite book|author=Richard Serfozo|title=Basics of Applied Stochastic Processes|url=https://books.google.com/books?id=JBBRiuxTN0QC|year=2009|publisher=Springer Science & Business Media|isbn=978-3-540-89332-5|page=2}}</ref><ref name="Rozanov2012page58">{{cite book|author=Y.A. Rozanov|title=Markov Random Fields|url=https://books.google.com/books?id=wGUECAAAQBAJ|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4613-8190-7|page=58}}</ref>

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{ columns-list | colwidth = 30em |

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