更改

<br>

<br>

==Further examples更多示例==

==更多示例==

{{Main|Markov process}}

马尔可夫过程是一类重要的随机过程，在许多领域有着广泛的应用。<ref name="LatoucheRamaswami1999"/><ref name="KarlinTaylor2012page47">{{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=47}}</ref>例如，它们是一种称为[[Markov chain Monte Carlo]]的一般随机模拟方法的基础，该方法用于模拟具有特定概率分布的随机对象，并在贝叶斯统计中得到应用。<ref name="RubinsteinKroese2011page225">{{cite book|author1=Reuven Y. Rubinstein|author2=Dirk P. Kroese|title=Simulation and the Monte Carlo Method|url=https://books.google.com/books?id=yWcvT80gQK4C|year=2011|publisher=John Wiley & Sons|isbn=978-1-118-21052-9|page=225}}</ref><ref name="GamermanLopes2006">{{cite book|author1=Dani Gamerman|author2=Hedibert F. Lopes|title=Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Second Edition|url=https://books.google.com/books?id=yPvECi_L3bwC|year=2006|publisher=CRC Press|isbn=978-1-58488-587-0}}</ref>

{{Main |马尔可夫过程}}

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.

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>

马尔可夫特性的概念最初是针对连续和离散时间的随机过程，但它也适用于其它指标集，如<math>n</math>维欧氏空间，这导致随机变量的集合被称为马尔可夫随机场。<ref name="Rozanov2012page61">{{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=61}}</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=27}}</ref><ref name="Bremaud2013page253">{{cite book|author=Pierre Bremaud|title=Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues|url=https://books.google.com/books?id=jrPVBwAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4757-3124-8|page=253}}</ref>

The Brownian motion process and the Poisson process (in one dimension) are both examples of Markov processes<ref name="Ross1996page235and358">{{cite book|author=Sheldon M. Ross|title=Stochastic processes|url=https://books.google.com/books?id=ImUPAQAAMAAJ|year=1996|publisher=Wiley|isbn=978-0-471-12062-9|pages=235, 358}}</ref> in continuous time, while [[random walk]]s on the integers and the [[gambler's ruin]] problem are examples of Markov processes in discrete time.<ref name="Florescu2014page373">{{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=373, 374}}</ref><ref name="KarlinTaylor2012page49">{{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=49}}</ref>

===鞅 Martingale===

布朗运动过程和泊松过程（一维）都是马尔可夫过程的例子<ref name="Ross1996page235and358">{{cite book|author=Sheldon M. Ross|title=Stochastic processes|url=https://books.google.com/books？id=ImUPAQAAMAAJ |年=1996 | publisher=Wiley | isbn=978-0-471-12062-9 | pages=235358}</ref>，整数上的[[随机游走]]和[[赌徒破产]]问题是离散时间中马尔可夫过程的例子<ref name="Florescu2014page373">{{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=373，374}</ref><ref name=“KarlinTaylor2012page49”>{cite book | author1=Samuel Karlin | author2=Howard E.Taylor | title=随机过程第一门课程|网址=https://books.google.com/books？id=dSDxjX9nmmMC |年份=2012 | publisher=学术出版社| isbn=978-0-08-057041-9 | page=49}</ref>

'''鞅 Martingale'''是一个离散时间或连续时间的随机过程，其性质是在给定过程的当前值和所有过去值的情况下，每个未来值的条件期望值等于当前值。在离散时间中，如果此属性适用于下一个值，则它适用于所有未来值。鞅的精确数学定义需要另外两个条件与过滤的数学概念相结合，这与随时间推移增加可用信息的直觉有关。鞅通常被定义为实值，<ref name="Klebaner2005page65">{{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=65}}</ref><ref name="KaratzasShreve2014page11">{{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=11}}</ref><ref name="Williams1991page93">{{cite book|author=David Williams|title=Probability with Martingales|url=https://books.google.com/books?id=e9saZ0YSi-AC|year=1991|publisher=Cambridge University Press|isbn=978-0-521-40605-5|pages=93, 94}}</ref> but they can also be complex-valued<ref name="Doob1990page292">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=NrsrAAAAYAAJ|year=1990|publisher=Wiley|pages=292, 293}}</ref>或更一般的。<ref name="Pisier2016">{{cite book|author=Gilles Pisier|title=Martingales in Banach Spaces|url=https://books.google.com/books?id=n3JNDAAAQBAJ&pg=PR4|year=2016|publisher=Cambridge University Press|isbn=978-1-316-67946-3}}</ref>

A Markov chain is a type of Markov process that has either discrete [[state space]] or discrete index set (often representing time), but the precise definition of a Markov chain varies.<ref name="Asmussen2003page7">{{cite book|url=https://books.google.com/books?id=BeYaTxesKy0C|title=Applied Probability and Queues|year=2003|publisher=Springer Science & Business Media|isbn=978-0-387-00211-8|page=7|author=Søren Asmussen}}</ref> For example, it is common to define a Markov chain as a Markov process in either [[Continuous and discrete variables|discrete or continuous time]] with a countable state space (thus regardless of the nature of time),<ref name="Parzen1999page188">{{cite book|url=https://books.google.com/books?id=0mB2CQAAQBAJ|title=Stochastic Processes|year=2015|publisher=Courier Dover Publications|isbn=978-0-486-79688-8|page=188|author=Emanuel Parzen}}</ref><ref name="KarlinTaylor2012page29">{{cite book|url=https://books.google.com/books?id=dSDxjX9nmmMC|title=A First Course in Stochastic Processes|year=2012|publisher=Academic Press|isbn=978-0-08-057041-9|pages=29, 30|author1=Samuel Karlin|author2=Howard E. Taylor}}</ref><ref name="Lamperti1977chap6">{{cite book|url=https://books.google.com/books?id=Pd4cvgAACAAJ|title=Stochastic processes: a survey of the mathematical theory|publisher=Springer-Verlag|year=1977|isbn=978-3-540-90275-1|pages=106–121|author=John Lamperti}}</ref><ref name="Ross1996page174and231">{{cite book|url=https://books.google.com/books?id=ImUPAQAAMAAJ|title=Stochastic processes|publisher=Wiley|year=1996|isbn=978-0-471-12062-9|pages=174, 231|author=Sheldon M. Ross}}</ref> but it has been also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space).<ref name="Asmussen2003page7" /> It has been argued that the first definition of a Markov chain, where it has discrete time, now tends to be used, despite the second definition having been used by researchers like [[Joseph Doob]] and [[Kai Lai Chung]].<ref name="MeynTweedie2009">{{cite book|author1=Sean Meyn|author2=Richard L. Tweedie|title=Markov Chains and Stochastic Stability|url=https://books.google.com/books?id=Md7RnYEPkJwC|year=2009|publisher=Cambridge University Press|isbn=978-0-521-73182-9|page=19}}</ref>

对称随机游动和Wiener过程（具有零漂移）分别是离散时间和连续时间的鞅的例子。<ref name="Klebaner2005page65"/><ref name="KaratzasShreve2014page11"/>对于一个[[独立且同分布]]随机变量的[[序列]]<math>X_1, X_2, X_3, \dots</math>且平均值为零，由连续部分和<math>X_1,X_1+ X_2, X_1+ X_2+X_3, \dots</math> 构成的随机过程是一个离散时间鞅<ref name="Steele2012page12">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|pages=12, 13}}</ref>，离散时间鞅推广了独立随机变量的部分和的概念。<ref name="HallHeyde2014page2">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year=2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|page=2}}</ref>

Markov processes form an important class of stochastic processes and have applications in many areas.<ref name="LatoucheRamaswami1999"/><ref name="KarlinTaylor2012page47">{{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=47}}</ref> For example, they are the basis for a general stochastic simulation method known as [[Markov chain Monte Carlo]], which is used for simulating random objects with specific probability distributions, and has found application in [[Bayesian statistics]].<ref name="RubinsteinKroese2011page225">{{cite book|author1=Reuven Y. Rubinstein|author2=Dirk P. Kroese|title=Simulation and the Monte Carlo Method|url=https://books.google.com/books?id=yWcvT80gQK4C|year=2011|publisher=John Wiley & Sons|isbn=978-1-118-21052-9|page=225}}</ref><ref name="GamermanLopes2006">{{cite book|author1=Dani Gamerman|author2=Hedibert F. Lopes|title=Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Second Edition|url=https://books.google.com/books?id=yPvECi_L3bwC|year=2006|publisher=CRC Press|isbn=978-1-58488-587-0}}</ref>

数学上的鞅形式化了公平博弈的概念，<ref name="Ross1996page295">{{cite book|author=Sheldon M. Ross|title=Stochastic processes|url=https://books.google.com/books?id=ImUPAQAAMAAJ|year=1996|publisher=Wiley|isbn=978-0-471-12062-9|page=295}}</ref>它们最初的开发目的是表明不可能赢得一场公平的比赛。<ref name="Steele2012page11"/>但现在它们被用于许多概率领域，这是研究它们的主要原因之一。<ref name="Williams1991page93"/><ref name="Steele2012page11">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=11}}</ref><ref name="Kallenberg2002page96">{{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=96}}</ref>许多概率问题已经通过在问题中找到鞅并加以研究而得到解决。<ref name="Steele2012page371">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=371}}</ref>在给定鞅矩的条件下，鞅会收敛，因此经常使用鞅得到收敛结果，这主要是由于[[鞅收敛定理]]s。<ref name="HallHeyde2014page2"/><ref name="Steele2012page22">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=22}}</ref><ref name="GrimmettStirzaker2001page336">{{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=336}}</ref>

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.<ref name="Rozanov2012page61">{{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=61}}</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=27}}</ref><ref name="Bremaud2013page253">{{cite book|author=Pierre Bremaud|title=Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues|url=https://books.google.com/books?id=jrPVBwAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4757-3124-8|page=253}}</ref>

鞅在统计学中有许多应用，但有人指出，它的使用和应用并不像它在统计学领域那样广泛，尤其是统计推断。<ref name="GlassermanKou2006">{{cite journal|last1=Glasserman|first1=Paul|last2=Kou|first2=Steven|title=A Conversation with Chris Heyde|journal=Statistical Science|volume=21|issue=2|year=2006|pages=292, 293|issn=0883-4237|doi=10.1214/088342306000000088|arxiv=math/0609294|bibcode=2006math......9294G}}</ref>他们在排队论和棕榈微积分等概率论领域找到了应用<ref name="BaccelliBremaud2013">{{cite book|author1=Francois Baccelli|author2=Pierre Bremaud|title=Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences|url=https://books.google.com/books?id=DH3pCAAAQBAJ&pg=PR2|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-11657-9}}</ref>以及其他领域，如经济学。<ref name="HallHeyde2014pageX">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year= 2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|page=x}}</ref> and finance.<ref name="MusielaRutkowski2006"/>

==='''鞅Martingale'''===

===莱维过程 Lévy process===

 

'''莱维过程'''是随机过程的一种类型，可以看作是连续时间中随机游动的推广。<ref name="Applebaum2004page1337"/><ref name="Bertoin1998pageVIII">{{cite book|author=Jean Bertoin|title=Lévy Processes|url=https://books.google.com/books?id=ftcsQgMp5cUC&pg=PR8|year=1998|publisher=Cambridge University Press|isbn=978-0-521-64632-1|page=viii}}</ref>这些过程在金融、流体力学等领域有着广泛的应用。<ref name="Applebaum2004page1336">{{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=1336}}</ref><ref name="ApplebaumBook2004page69">{{cite book|author=David Applebaum|title=Lévy Processes and Stochastic Calculus|url=https://books.google.com/books?id=q7eDUjdJxIkC|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83263-2|page=69}}</ref>这些过程和过程的独立性被称为平稳过程的主要特征。换句话说，一个随机过程<math>X</math>是一个Lévy过程，如果对非负数<math>n</math>，<math>0\leq t_1\leq \dots \leq t_n</math>，当<math>n-1</math>递增

 

A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value. In discrete time, if this property holds for the next value, then it holds for all future values. The exact mathematical definition of a martingale requires two other conditions coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes. Martingales are usually defined to be real-valued,<ref name="Klebaner2005page65">{{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=65}}</ref><ref name="KaratzasShreve2014page11">{{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=11}}</ref><ref name="Williams1991page93">{{cite book|author=David Williams|title=Probability with Martingales|url=https://books.google.com/books?id=e9saZ0YSi-AC|year=1991|publisher=Cambridge University Press|isbn=978-0-521-40605-5|pages=93, 94}}</ref> but they can also be complex-valued<ref name="Doob1990page292">{{cite book|author=Joseph L. Doob|title=Stochastic processes|url=https://books.google.com/books?id=NrsrAAAAYAAJ|year=1990|publisher=Wiley|pages=292, 293}}</ref> or even more general.<ref name="Pisier2016">{{cite book|author=Gilles Pisier|title=Martingales in Banach Spaces|url=https://books.google.com/books?id=n3JNDAAAQBAJ&pg=PR4|year=2016|publisher=Cambridge University Press|isbn=978-1-316-67946-3}}</ref>

 

'''鞅Martingale'''是一个离散时间或连续时间的随机过程，其性质是在给定过程的当前值和所有过去值的情况下，每个未来值的条件期望值等于当前值。在离散时间中，如果此属性适用于下一个值，则它适用于所有未来值。'''鞅Martingale'''的精确数学定义需要另外两个条件与过滤的数学概念相结合，这与随时间推移增加可用信息的直觉有关。'''鞅Martingale'''通常被定义为实值，<ref name=“Klebaner2005page65”>{cite book | author=Fima C.Klebaner | title=随机微积分及其应用简介=https://books.google.com/books？id=JYzW0uqQxB0C | year=2005 | publisher=Imperial College Press | isbn=978-1-86094-555-7 | page=65}</ref><ref name=“KaratzasShreve2014page11”>{cite book | author1=Ioannis Karatzas | author2=Steven Shreve | title=布朗运动和随机微积分| url=https://books.google.com/books？id=w0SgBQAAQBAJ&pg=PT5 | year=1991 | publisher=Springer | isbn=978-1-4612-0949-2 | page=11}</ref><ref name=“Williams1991page93”>{引用图书|作者=David Williams | title=Probability with鞅| url=https://books.google.com/books？id=e9saZ0YSi AC | year=1991 | publisher=Cambridge University Press | isbn=978-0-521-40605-5 | pages=93，94}</ref>但是它们也可以是复杂值<ref name=“Doob1990page292”>{cite book | author=Joseph L.Doob | title=randouses | url=https://books.google.com/books？id=nrsraaayaaj | year=1990 | publisher=Wiley | pages=2929293}</ref>或更一般的。<ref name=“Pisier2016”>{cite book | author=Gilles Pisier | title=Banach空格中的鞅| url=https://books.google.com/books？id=n3JNDAAAQBAJ&pg=PR4 | year=2016 | publisher=Cambridge University Press | isbn=978-1-316-67946-3}</ref>

 

 

 

 

 

 

通过应用适当的变换，也可以从随机过程中产生'''鞅Martingale'''，这就是齐次泊松过程（在实线上）的情形，其结果是一个称为“补偿泊松过程”的鞅。<ref name=“karatzashreserve2014page11”/>也可以从其他鞅中构建鞅。<ref name=“Steele2012page12”/>例如，有基于鞅的鞅Wiener过程，形成连续时间鞅。<ref name=“Klebaner2005page65”/><ref name=“Steele2012page115”>{cite book | author=J.Michael Steele | title=随机微积分与金融应用=https://books.google.com/books？id=fsgkbaaqbaj&pg=PR4 | year=2012 | publisher=Springer科学与商业媒体| isbn=978-1-4684-9305-4 | page=115}</ref>

 

 

 

 

 

 

 

Lévy processes are types of stochastic processes that can be considered as generalizations of random walks in continuous time.<ref name="Applebaum2004page1337"/><ref name="Bertoin1998pageVIII">{{cite book|author=Jean Bertoin|title=Lévy Processes|url=https://books.google.com/books?id=ftcsQgMp5cUC&pg=PR8|year=1998|publisher=Cambridge University Press|isbn=978-0-521-64632-1|page=viii}}</ref> These processes have many applications in fields such as finance, fluid mechanics, physics and biology.<ref name="Applebaum2004page1336">{{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=1336}}</ref><ref name="ApplebaumBook2004page69">{{cite book|author=David Applebaum|title=Lévy Processes and Stochastic Calculus|url=https://books.google.com/books?id=q7eDUjdJxIkC|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83263-2|page=69}}</ref> 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

 

''' 莱维Lévy过程'''是随机过程的一种类型，可以看作是连续时间中随机游动的推广<ref name=“Applebaum2004page1337”/><ref name=“Bertoin1998pageVIII”>{引用图书|作者=Jean Bertoin | title=莱维过程 |网址=https://books.google.com/books？id=ftcsQgMp5cUC&pg=PR8 | year=1998 | publisher=Cambridge University Press | isbn=978-0-521-64632-1 | page=viii}}</ref>这些过程在金融、流体力学等领域有着广泛的应用，<ref name="Applebaum2004page1336">{{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=1336}}</ref><ref name="ApplebaumBook2004page69">{{cite book|author=David Applebaum|title=Lévy Processes and Stochastic Calculus|url=https://books.google.com/books?id=q7eDUjdJxIkC|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83263-2|page=69}}</ref> 这些过程和过程的独立性被称为平稳过程的主要特征。换句话说，一个随机过程<math>X</math>是一个Lévy过程，如果对非负数<math>n</math>，<math>0\leq t_1\leq \dots \leq t_n</math>，当<math>n-1</math>递增

<center><math>

<center><math>

</math></center>

</math></center>

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它们彼此独立，每个增量的分布只取决于时间的差异。<ref name="Applebaum2004page1337"/>

点过程是随机分布在某些数学空间（如实线、<math>n</math>维欧几里德空间或更抽象的空间）上的点的集合。有时“点过程”一词并不可取，因为历史上“过程”一词表示某个系统在时间上的演变，因此，点过程也被称为“随机点域”。<ref name=“ChiuStoyan2013page109”>{cite book | author1=Sung Nok Chiu | author2=Dietrich Stoyan | author3=Wilfrid S.Kendall | author4=Joseph Mecke | title=随机几何及其应用| url=https://books.google.com/books？id=825NfM6Nc EC | year=2013 | publisher=John Wiley&Sons | isbn=978-1-118-65825-3 | page=109}</ref>点过程有不同的解释，这样一个随机计数度量或随机集。<ref name=“ChiuStoyan2013page108”>{cite book | author1=Sung Nok Chiu | author2=Dietrich Stoyan | author3=Wilfrid S.Kendall | author4=Joseph Mecke | title=随机几何及其应用=https://books.google.com/books？id=825NfM6Nc EC |年份=2013 | publisher=John Wiley&Sons | isbn=978-1-118-65825-3 | page=108}</ref><ref name=“Haenggi2013page10”>{cite book |作者=Martin Haenggi | title=无线网络的随机几何| url=https://books.google.com/books？id=CLtDhblwWEgC | year=2013 | publisher=Cambridge University Press | isbn=978-1-107-01469-5 | page=10}</ref>一些作者将点过程和随机过程视为两个不同的对象，因此点过程是随机过程产生或与随机过程相关联的随机对象，<ref name=“daleyviere-Jones 2006page194”>{cite book | author1=D.J.Daley | author2=D.Vere Jones | title=点过程理论导论：第一卷：基本理论与方法|网址=https://books.google.com/books？id=6Sv4BwAAQBAJ | year=2006 | publisher=Springer Science&Business Media | isbn=978-0-387-21564-8 | page=194}</ref><ref name=“CoxIsham1980page3”>{引用图书| author1=D.R.Cox | author2=Valerie Isham | title=Point Processes | url=https://books.google.com/books？id=KWF2xY6s3PoC | year=1980 | publisher=CRC Press | isbn=978-0-412-21910-8 | page=3}</ref>尽管已经注意到点过程和随机过程之间的区别并不清楚。<ref name=“CoxIsham1980page3”/>

===随机 Random field===

or article citation details via DOI numbers:

随机场是由一个<math>n</math>维欧几里德空间或流形索引的随机变量的集合。一般来说，随机场可以看作是随机过程的一个例子，其中，索引集不一定是实行的子集。<ref name="AdlerTaylor2009page7"/>但是有一个约定，当索引具有两个或多个维度时，随机变量的索引集合称为随机字段。<ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/><ref name="KoralovSinai2007page171">{{cite book|author1=Leonid Koralov|author2=Yakov G. Sinai|title=Theory of Probability and Random Processes|url=https://books.google.com/books?id=tlWOphOFRgwC|year=2007|publisher=Springer Science & Business Media|isbn=978-3-540-68829-7|page=171}}</ref>如果随机过程的具体定义要求索引集是实线的子集，那么随机场可以看作是随机过程的一个推广。<ref name="ApplebaumBook2004page19">{{cite book|author=David Applebaum|title=Lévy Processes and Stochastic Calculus|url=https://books.google.com/books?id=q7eDUjdJxIkC|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83263-2|page=19}}</ref>

Other authors consider a point process as a stochastic process, where the process is indexed by sets of the underlying space{{efn|In the context of point processes, the term "state space" can mean the space on which the point process is defined such as the real line,<ref name="Kingman1992page8">{{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=8}}</ref><ref name="MollerWaagepetersen2003page7">{{cite book|author1=Jesper Moller|author2=Rasmus Plenge Waagepetersen|title=Statistical Inference and Simulation for Spatial Point Processes|url=https://books.google.com/books?id=dBNOHvElXZ4C|year=2003|publisher=CRC Press|isbn=978-0-203-49693-0|page=7}}</ref> 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.<ref name="KarlinTaylor2012page31">{{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=31}}</ref><ref name="Schmidt2014page99">{{cite book|author=Volker Schmidt|title=Stochastic Geometry, Spatial Statistics and Random Fields: Models and Algorithms|url=https://books.google.com/books?id=brsUBQAAQBAJ&pg=PR5|date= 2014|publisher=Springer|isbn=978-3-319-10064-7|page=99}}</ref> Other stochastic processes such as renewal and counting processes are studied in the theory of point processes.<ref name="DaleyVere-Jones200">{{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}}</ref><ref name="CoxIsham1980">{{cite book|author1=D.R. Cox|author2=Valerie Isham|title=Point Processes|url=https://books.google.com/books?id=KWF2xY6s3PoC|year=1980|publisher=CRC Press|isbn=978-0-412-21910-8}}</ref>

===点过程 Point process===

另一些作者认为点过程是一个随机过程，其中过程由一组底层空间{efn |索引；在点过程的上下文中，“状态空间”一词可以指定义点过程的空间，如实线，<ref name=“kingmann1992page8”>{cite book | author=J.F.C.Kingman | title=Poisson Processes |网址=https://books.google.com/books？id=VEiM OtwDHkC | year=1992 | publisher=Clarendon Press | isbn=978-0-19-159124-2 | page=8}</ref><ref name=“MollerWaagepetersen2003page7”>{cite book | author1=Jesper Moller | author2=Rasmus-Plenge Waagepetersen | title=空间点过程的统计推断和模拟| url=https://books.google.com/books？id=dBNOHvElXZ4C | year=2003 | publisher=CRCRC Press | isbn=978-0-203-49693-0 | page=7}</ref>其中与随机过程术语中的指标集相对应的指标集。}}其上定义它的地方，如实线或<数学>n</math>n</math>-维维的欧几里得空间。<ref name=“KarlintayRo2012Page31”>{本书|124;author1=Samuel Karlin | author2=Howard E Howard E.Howard E.Taylor | title=title=title=A随机过程第一课程|网址=https://books.google.com/books？id=dSDxjX9nmmMC | year=2012 | publisher=academical Press | isbn=978-0-08-057041-9 | page=31}</ref><ref name=“Schmidt2014page99”>{cite book | author=Volker-Schmidt | title=随机几何、空间统计和随机场：模型和算法| url=https://books.google.com/books？id=brsUBQAAQBAJ&pg=PR5 | date=2014 | publisher=Springer | isbn=978-3-319-10064-7 | page=99}</ref>其他随机过程，如更新和计数过程，在点过程理论中进行了研究一、基本理论与方法|网址=https://books.google.com/books？id=6Sv4BwAAQBAJ | year=2006 | publisher=Springer Science&Business Media | isbn=978-0-387-21564-8}</ref><ref name=“CoxIsham1980”>{cite book | author1=D.R.Cox | author2=Valerie Isham | title=Point processs|网址=https://books.google.com/books？id=KWF2xY6s3PoC |年=1980 |发行人=CRC出版社| isbn=978-0-412-21910-8}</ref>

点过程是随机分布在某些数学空间（如实线、<math>n</math>维欧几里德空间或更抽象的空间）上的点的集合。有时“点过程”一词并不可取，因为历史上“过程”一词表示某个系统在时间上的演变，因此，点过程也被称为“随机点域”。<ref name="ChiuStoyan2013page109">{{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|page=109}}</ref>>点过程有不同的解释，这样一个随机计数度量或随机集。<ref name="ChiuStoyan2013page108">{{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|page=108}}</ref><ref name="Haenggi2013page10">{{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|page=10}}</ref>一些作者将点过程和随机过程视为两个不同的对象，因此点过程是随机过程产生或与随机过程相关联的随机对象，<ref name="DaleyVere-Jones2006page194">{{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=194}}</ref><ref name="CoxIsham1980page3">{{cite book|author1=D.R. Cox|author2=Valerie Isham|title=Point Processes|url=https://books.google.com/books?id=KWF2xY6s3PoC|year=1980|publisher=CRC Press|isbn=978-0-412-21910-8|page=3}}</ref>尽管已经注意到点过程和随机过程之间的区别并不清楚。<ref name="CoxIsham1980page3"/>

http://reftag.appspot.com/doiweb.py

Other authors consider a point process as a stochastic process, where the process is indexed by sets of the underlying space{{efn|In the context of point processes, the term "state space" can mean the space on which the point process is defined such as the real line, which corresponds to the index set in stochastic process terminology.}} 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.

Http://reftag.appspot.com/doiweb.py

另一些作者认为点过程是一个随机过程，其中过程由一组底层空间（在点过程的上下文中，“状态空间”一词可以指定义点过程的空间，如实线，<ref name="Kingman1992page8">{{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=8}}</ref><ref name="MollerWaagepetersen2003page7">{{cite book|author1=Jesper Moller|author2=Rasmus Plenge Waagepetersen|title=Statistical Inference and Simulation for Spatial Point Processes|url=https://books.google.com/books?id=dBNOHvElXZ4C|year=2003|publisher=CRC Press|isbn=978-0-203-49693-0|page=7}}</ref>其中与随机过程术语中的指标集相对应的指标集。）其上定义它的地方，如实线或<math>n</math>-维的欧几里得空间。<ref name="KarlinTaylor2012page31">{{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=31}}</ref><ref name="Schmidt2014page99">{{cite book|author=Volker Schmidt|title=Stochastic Geometry, Spatial Statistics and Random Fields: Models and Algorithms|url=https://books.google.com/books?id=brsUBQAAQBAJ&pg=PR5|date= 2014|publisher=Springer|isbn=978-3-319-10064-7|page=99}}</ref>其他随机过程，如更新和计数过程，在点过程理论中进行了研究一、基本理论与方法。<ref name="DaleyVere-Jones200">{{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}}</ref><ref name="CoxIsham1980">{{cite book|author1=D.R. Cox|author2=Valerie Isham|title=Point Processes|url=https://books.google.com/books?id=KWF2xY6s3PoC|year=1980|publisher=CRC Press|isbn=978-0-412-21910-8}}</ref>

有关其他来源，请参阅: WP: CITET

继Cardano之后，[[雅各布·伯努利]] Jakob Bernoulli（也被称为James Bernoulli或Jacques Bernoulli<ref name="Hald2005page221">{{cite book|author=Anders Hald|title=A History of Probability and Statistics and Their Applications before 1750|url=https://books.google.com/books?id=pOQy6-qnVx8C|year=2005|publisher=John Wiley & Sons|isbn=978-0-471-72517-6|page=221}}</ref>）写了[[魔术师Ars conjuctandi]]，在概率论史上被认为是重大事件。<ref name=":1" />伯努利的书出版于1713年，也是在他死后出版的，激发了许多数学家研究概率。<ref name=":1" /><ref name="Maistrov2014page56">{{cite book|author=L. E. Maistrov|title=Probability Theory: A Historical Sketch|url=https://books.google.com/books?id=2ZbiBQAAQBAJ&pg=PR9|year=2014|publisher=Elsevier Science|isbn=978-1-4832-1863-2|page=56}}</ref><ref name="Tabak2014page37">{{cite book|author=John Tabak|title=Probability and Statistics: The Science of Uncertainty|url=https://books.google.com/books?id=h3WVqBPHboAC|year=2014|publisher=Infobase Publishing|isbn=978-0-8160-6873-9|page=37}}</ref>

Probability theory has its origins in games of chance, which have a long history, with some games being played thousands of years ago,<ref name=":1">{{Cite book|title=Markov Chains: From Theory to Implementation and Experimentation|last=Gagniuc|first=Paul A.|publisher=John Wiley & Sons|year=2017|isbn=978-1-119-38755-8|location=US|pages=1–2}}</ref><ref name="David1955">{{cite journal|last1=David|first1=F. N.|title=Studies in the History of Probability and Statistics I. Dicing and Gaming (A Note on the History of Probability)|journal=Biometrika|volume=42|issue=1/2|pages=1–15|year=1955|issn=0006-3444|doi=10.2307/2333419|jstor=2333419}}</ref> but very little analysis on them was done in terms of probability.<ref name=":1" /><ref name="Maistrov2014page1">{{cite book|author=L. E. Maistrov|title=Probability Theory: A Historical Sketch|url=https://books.google.com/books?id=2ZbiBQAAQBAJ&pg=PR9|year=2014|publisher=Elsevier Science|isbn=978-1-4832-1863-2|page=1}}</ref> The year 1654 is often considered the birth of probability theory when French mathematicians [[Pierre Fermat]] and [[Blaise Pascal]] had a written correspondence on probability, motivated by a [[Problem of points|gambling problem]].<ref name=":1" /><ref name="Seneta2006page1">{{cite book|last1=Seneta|first1=E.|title=Encyclopedia of Statistical Sciences|chapter=Probability, History of|year=2006|doi=10.1002/0471667196.ess2065.pub2|page=1|isbn=978-0471667193}}</ref><ref name="Tabak2014page24to26">{{cite book|author=John Tabak|title=Probability and Statistics: The Science of Uncertainty|url=https://books.google.com/books?id=h3WVqBPHboAC|year=2014|publisher=Infobase Publishing|isbn=978-0-8160-6873-9|pages=24–26}}</ref> But there was earlier mathematical work done on the probability of gambling games such as ''Liber de Ludo Aleae'' by [[Gerolamo Cardano]], written in the 16th century but posthumously published later in 1663.<ref name=":1" /><ref name="Bellhouse2005">{{cite journal|last1=Bellhouse|first1=David|title=Decoding Cardano's Liber de Ludo Aleae|journal=Historia Mathematica|volume=32|issue=2|year=2005|pages=180–202|issn=0315-0860|doi=10.1016/j.hm.2004.04.001|doi-access=free}}</ref>

===统计力学 Statistical mechanics===

概率论起源于机会博弈，它有着悠久的历史，有些游戏在几千年前就已经开始了，{{引文图书〈title=马尔可夫链：从理论到实施和实验| last=Gagniuc | first=Paul A.；publisher=John Wiley&Sons | year=2017 | isbn=978-1-119-38755-8 |地理位置=美国|页面=1-1–2}}</ref><ref name=“David1955”{{引用杂志| last1=David | first1=F.N.| title=在研究中的研究在研究中的研究在研究中的研究中的研究|位置=位置=美国124;位置=美国概率统计史I.划片游戏和游戏（概率历史注注）| journal=Biometrika |volume=42 | issue=1/2 | pages=1-15年；year=1955 | issn=0006-3444；doi=10.2307/2333419；jstor=2333419}}</ref>但很少从概率的角度对其进行分析。<ref name=“：1”/><ref name=“Maistorv2014Page1”>{{ci书| ci书124; author=L.E.Maistorv Maistorv Maistorv Maistorv Maistorv Maistorv Maistorv Ma|标题=概率论：A历史素描|网址=https://books.google.com/books？id=2ZbiBQAAQBAJ&pg=PR9 | year=2014 | publisher=Elsevier Science | isbn=978-1-4832-1863-2 | page=1}</ref>当法国数学家[[Pierre Fermat]]和[[Blaise Pascal]]在概率论上有过书面通信时，1654年通常被认为是概率论的诞生，受[[点数问题|赌博问题].<ref name=“：1”/><ref name=“Seneta2006page1”>{cite book | last1=Seneta | first1=E.| title=统计科学百科全书| chapter=Probability，历史|年=2006 | doi=10.1002/0471667196.ess2065.pub2 | page=1 | isbn=978-0471667193}</ref><ref name=“Tabak2014page24to26”>{cite book | author=John Tabak | title=Probability and Statistics:不确定性科学| url=https://books.google.com/books？id=h3WVqBPHboAC | year=2014 | publisher=Infobase Publishing | isbn=978-0-8160-6873-9 | pages=24–26}</ref>但是早期有关于赌博游戏概率的数学研究，比如[[Gerolamo Cardano]]的“Liber de Ludo Aleae”，16世纪写于16世纪，死后于1663年发表。<ref name=“：1“/><ref name=“bellhous2005”>{〈cite journal | last1=Bellhouse | first1=David；title=解码Cardano's Liberde Ludo Aleaeaeae | journal=Historica Mathematica | volume=32；issue=2；year=2005 | Pages180–202 | issn=0315-0860 | doi=10.1016/j.hm 2004.2004.04.001 | jo180–202 | issn=10.202 doiaccess=free}}</ref>

在物理科学中，科学家们在19世纪发展了[[统计力学]]这门学科，在这个学科中，物理系统，例如装满气体的容器，可以从数学上看作或处理为许多运动粒子的集合。尽管有些科学家试图将随机性纳入统计物理学，比如[[鲁道夫·克劳修斯]]，大部分工作没有或几乎没有随机性。<ref name="Truesdell1975page22">{{cite journal|last1=Truesdell|first1=C.|title=Early kinetic theories of gases|journal=Archive for History of Exact Sciences|volume=15|issue=1|year=1975|pages=22–23|issn=0003-9519|doi=10.1007/BF00327232}}</ref><ref name="Brush1967page150">{{cite journal|last1=Brush|first1=Stephen G.|title=Foundations of statistical mechanics 1845?1915|journal=Archive for History of Exact Sciences|volume=4|issue=3|year=1967|pages=150–151|issn=0003-9519|doi=10.1007/BF00412958}}</ref>

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这种情况在1859年发生了变化，当时[[詹姆斯·克拉克·麦克斯韦]] James Clerk Maxwell对该领域做出了重大贡献，更具体地说，是对气体动力学理论的贡献，通过介绍他的工作，他假设气体粒子以随机速度随机方向移动<ref name="Truesdell1975page31">{{cite journal|last1=Truesdell|first1=C.|title=Early kinetic theories of gases|journal=Archive for History of Exact Sciences|volume=15|issue=1|year=1975|pages=31–32|issn=0003-9519|doi=10.1007/BF00327232}}</ref><ref name="Brush1958">{{cite journal|last1=Brush|first1=S.G.|title=The development of the kinetic theory of gases IV. Maxwell|journal=Annals of Science|volume=14|issue=4|year=1958|pages=243–255|issn=0003-3790|doi=10.1080/00033795800200147}}</ref>气体动力学理论和统计物理在19世纪后半叶继续发展，主要由克劳修斯，[[路德维希·玻尔兹曼]]和[[约西亚·威拉德·吉布斯]]完成，这项工作后来对[[阿尔伯特·爱因斯坦]]关于[[布朗运动]]的数学模型产生了影响。<ref name="Brush1968page15">{{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=15–16|issn=0003-9519|doi=10.1007/BF00328110}}</ref>

After Cardano, [[Jakob Bernoulli]]{{efn|Also known as James or Jacques Bernoulli.<ref name="Hald2005page221">{{cite book|author=Anders Hald|title=A History of Probability and Statistics and Their Applications before 1750|url=https://books.google.com/books?id=pOQy6-qnVx8C|year=2005|publisher=John Wiley & Sons|isbn=978-0-471-72517-6|page=221}}</ref>}} wrote [[Ars Conjectandi]], which is considered a significant event in the history of probability theory.<ref name=":1" /> Bernoulli's book was published, also posthumously, in 1713 and inspired many mathematicians to study probability.<ref name=":1" /><ref name="Maistrov2014page56">{{cite book|author=L. E. Maistrov|title=Probability Theory: A Historical Sketch|url=https://books.google.com/books?id=2ZbiBQAAQBAJ&pg=PR9|year=2014|publisher=Elsevier Science|isbn=978-1-4832-1863-2|page=56}}</ref><ref name="Tabak2014page37">{{cite book|author=John Tabak|title=Probability and Statistics: The Science of Uncertainty|url=https://books.google.com/books?id=h3WVqBPHboAC|year=2014|publisher=Infobase Publishing|isbn=978-0-8160-6873-9|page=37}}</ref> But despite some renowned mathematicians contributing to probability theory, such as [[Pierre-Simon Laplace]], [[Abraham de Moivre]], [[Carl Gauss]], [[Siméon Poisson]] and [[Pafnuty Chebyshev]],<ref name="Chung1998">{{cite journal|last1=Chung|first1=Kai Lai|title=Probability and Doob|journal=The American Mathematical Monthly|volume=105|issue=1|pages=28–35|year=1998|issn=0002-9890|doi=10.2307/2589523|jstor=2589523}}</ref><ref name="Bingham2000">{{cite journal|last1=Bingham|first1=N.|title=Studies in the history of probability and statistics XLVI. Measure into probability: from Lebesgue to Kolmogorov|journal=Biometrika|volume=87|issue=1|year=2000|pages=145–156|issn=0006-3444|doi=10.1093/biomet/87.1.145}}</ref> most of the mathematical community{{efn|It has been remarked that a notable exception was the St Petersburg School in Russia, where mathematicians  led by Chebyshev studied probability theory.<ref name="BenziBenzi2007">{{cite journal|last1=Benzi|first1=Margherita|last2=Benzi|first2=Michele|last3=Seneta|first3=Eugene|title=Francesco Paolo Cantelli. b. 20 December 1875 d. 21 July 1966|journal=International Statistical Review|volume=75|issue=2|year=2007|page=128|issn=0306-7734|doi=10.1111/j.1751-5823.2007.00009.x}}</ref>}} did not consider probability theory to be part of mathematics until the 20th century.<ref name="Chung1998"/><ref name="BenziBenzi2007"/><ref name="Doob1996">{{cite journal|last1=Doob|first1=Joseph L.|title=The Development of Rigor in Mathematical Probability (1900-1950)|journal=The American Mathematical Monthly|volume=103|issue=7|pages=586–595|year=1996|issn=0002-9890|doi=10.2307/2974673|jstor=2974673}}</ref><ref name="Cramer1976">{{cite journal|last1=Cramer|first1=Harald|title=Half a Century with Probability Theory: Some Personal Recollections|journal=The Annals of Probability|volume=4|issue=4|year=1976|pages=509–546|issn=0091-1798|doi=10.1214/aop/1176996025|doi-access=free}}</ref>

1900年在巴黎举行的国际数学家大会上，David Hilbert提出了一份[[Hilbert问题|数学问题]]的清单，其中他的第六个问题要求对涉及公理的物理和概率进行数学处理。<ref name="Bingham2000"/>大约在20世纪初，数学家发展了测量理论，这是研究数学函数积分的数学分支，其中两位创始人是法国数学家Henri Lebesgue和Émile Borel。1925年，另一位法国数学家Paul Lévy出版了第一本使用测度论思想的概率论书籍。<ref name="Bingham2000"/>

===Statistical mechanics统计力学===

20世纪20年代，苏联的数学家们对概率论做出了重大贡献，比如Sergei Bernstein，Aleksandr Khinchin，（Khinchin这个名字也用英语写成（或音译成）Khintchine。<ref name="Doob1934">{{cite journal|last1=Doob|first1=Joseph|title=Stochastic Processes and Statistics|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=20|issue=6|year=1934|pages=376–379|doi=10.1073/pnas.20.6.376|pmid=16587907|pmc=1076423|bibcode=1934PNAS...20..376D}}</ref>）和Andrei Kolmogorov。<ref name="Cramer1976"/> Kolmogorov于1984年发表了基于测量理论的数学基础的首次尝试。概率论的概率论。<ref name="KendallBatchelor1990page33">{{cite journal|last1=Kendall|first1=D. G.|last2=Batchelor|first2=G. K.|last3=Bingham|first3=N. H.|last4=Hayman|first4=W. K.|last5=Hyland|first5=J. M. E.|last6=Lorentz|first6=G. G.|last7=Moffatt|first7=H. K.|last8=Parry|first8=W.|last9=Razborov|first9=A. A.|last10=Robinson|first10=C. A.|last11=Whittle|first11=P.|title=Andrei Nikolaevich Kolmogorov (1903–1987)|journal=Bulletin of the London Mathematical Society|volume=22|issue=1|year=1990|page=33|issn=0024-6093|doi=10.1112/blms/22.1.31}}</ref>在20世纪30年代初，Khinchin和Kolmogorov在20世纪30年代初建立了概率研讨会，这些研讨会由研究者参加，如Eugene Slutsky等和Nikolai Smirnov，<ref name="Vere-Jones2006page1">{{cite book|last1=Vere-Jones|first1=David|title=Encyclopedia of Statistical Sciences|chapter=Khinchin, Aleksandr Yakovlevich|page=1|year=2006|doi=10.1002/0471667196.ess6027.pub2|isbn=978-0471667193}}</ref>还有Khinchin给出了第一个随机变量的数学定义，把随机过程作为以实数线索引的一组随机变量。<ref name="Doob1934"/><ref name="Vere-Jones2006page4">{{cite book|last1=Vere-Jones|first1=David|title=Encyclopedia of Statistical Sciences|chapter=Khinchin, Aleksandr Yakovlevich|page=4|year=2006|doi=10.1002/0471667196.ess6027.pub2|isbn=978-0471667193}}</ref>（Doob在引用Khinchin时，使用了“机会变量”这个词，它曾经是“随机变量”的替代词。<ref name="Snell2005">{{cite journal|last1=Snell|first1=J. Laurie|title=Obituary: Joseph Leonard Doob|journal=Journal of Applied Probability|volume=42|issue=1|year=2005|page=251|issn=0021-9002|doi=10.1239/jap/1110381384|doi-access=free}}</ref>）

在物理科学中，科学家们在19世纪发展了[[统计力学]]这门学科，在这个学科中，物理系统，例如装满气体的容器，可以从数学上看作或处理为许多运动粒子的集合。尽管有些科学家试图将随机性纳入统计物理学，比如[[Rudolf Clausius]]，大部分工作没有或几乎没有随机性。<ref name=“Truesdell1975page22”>{cite journal | last1=Truesdell|first1=C.| title=早期气体动力学理论| journal=精确科学史档案|卷=15 |问题=1 |年份=1975 |页=22–23 | issn=0003-9519 | doi=10.1007/BF00327232}}<ref name=“Brush1967page150”>{{cite journal | last1=Brush | first1=Stephen G.| title=统计力学基础1845？1915年|期刊=精确科学史档案|卷=4 |问题=3 |年=1967 |页=150–151 | issn=0003-9519 | doi=10.1007/BF00412958}}</ref>

===现代概率论的诞生 Birth of modern probability theory===

This changed in 1859 when [[James Clerk Maxwell]] contributed significantly to the field, more specifically, to the kinetic theory of gases, by presenting work where he assumed the gas particles move in random directions at random velocities.<ref name="Truesdell1975page31">{{cite journal|last1=Truesdell|first1=C.|title=Early kinetic theories of gases|journal=Archive for History of Exact Sciences|volume=15|issue=1|year=1975|pages=31–32|issn=0003-9519|doi=10.1007/BF00327232}}</ref><ref name="Brush1958">{{cite journal|last1=Brush|first1=S.G.|title=The development of the kinetic theory of gases IV. Maxwell|journal=Annals of Science|volume=14|issue=4|year=1958|pages=243–255|issn=0003-3790|doi=10.1080/00033795800200147}}</ref> The kinetic theory of gases and statistical physics continued to be developed in the second half of the 19th century, with work done chiefly by Clausius, [[Ludwig Boltzmann]] and [[Josiah Gibbs]], which would later have an influence on [[Albert Einstein]]'s mathematical model for [[Brownian movement]].<ref name="Brush1968page15">{{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=15–16|issn=0003-9519|doi=10.1007/BF00328110}}</ref>

1933年，Andrei Kolmogorov在德国出版了一本关于概率论基础的书，名为“概率计算的基本概念”，后来翻译成英文，1950年出版，作为概率论的基础。这本书的出版现在被广泛认为是现代概率论的诞生，当时概率论和随机过程理论成为数学的一部分。<ref name="Bingham2000"/><ref name="Cramer1976"/>

===Measure theory and probability theory测度论与概率论===

在Kolmogorov的书出版后，钦钦和科尔莫戈洛夫以及其他数学家如Joseph Doob、William Feller、Maurice Fréchet、Paul Lévy、Wolfgang Doeblin等对概率论和随机过程进行了进一步的基础性工作，和Harald Cramér。<ref name=“Bingham2000”/><ref name=“Cramer1976”/>

1900年在巴黎举行的[[国际数学家大会]]上，[[David Hilbert]]提出了一份[[Hilbert问题|数学问题]]的清单，其中他的第六个问题要求对涉及[[公理]]的物理和概率进行数学处理。<ref name=“Bingham2000”/>大约在20世纪初，数学家发展了测量理论，这是研究数学函数积分的数学分支，其中两位创始人是法国数学家[[Henri Lebesgue]]和[[Émile Borel]]。1925年，另一位法国数学家[[Paul Lévy（数学家）| Paul Lévy]]出版了第一本使用测度论思想的概率论书籍

几十年后，Cramér把20世纪30年代称为“数学概率论的英雄时期”。<ref name="Cramer1976"/>第二次世界大战极大地中断了概率论的发展，例如，Feller从瑞典迁移到美国。<ref name="Lindvall1991">{{cite journal|last1=Lindvall|first1=Torgny|title=W. Doeblin, 1915-1940|journal=The Annals of Probability|volume=19|issue=3|year=1991|pages=929–934|issn=0091-1798|doi=10.1214/aop/1176990329|doi-access=free}}</ref>

20世纪20年代，苏联的数学家们对概率论做出了重大贡献，比如[[Sergei Bernstein]]，[[Aleksandr Khinchin]]，{{efn | Khinchin这个名字也用英语写成（或音译成）Khintchine。<ref name=“Doob1934”>{cite journal | last1=Doob | first1=Joseph | title=随机过程与统计| journal=美国国家科学院学报美国体积＝20π＝6π＝376×379π＝10.1073 /PNAS 20.637 6pMID＝16587907πPMC＝1076423 BiBCODE＝1934 PNAS…20 .37 6D } </REF> }和[[Andrei Kolmogorov ] ] < 1929】命名为“CRAME1976”/Kolmogorov于1984年发表了基于测量理论的数学基础的首次尝试。概率论的概率论。<ref name=“KendallBatchelo1990 Page33”>{〈引用期刊| last1=Kendall | first1=D.G.| last2=Batchelor | first2=G.K.| last3=Bingham | first3=N.H.| last4=Hayman | first4=W.K.| last5=Hyland | first5=第一5=J.M.M.E.|124;最后6=洛伦兹|第一6=G.G.最后7=最后7=最后7=最后7=N.H.莫法特| first7=H.K.| last8=Parry | first8=W.| last9=Razborov | first9=A.A.| last10=Robinson | first10=C。A、 最后11=Whittle；first11=P.；title=AndreiNikolaevich Kolmogorov（1903-1987年）| journal=伦敦数学社会公报| volume=22 | issue=1 | year=1990年| page=33 | issn=0024-6093；doi=10.1112/blms/22.1.31}</ref>在20世纪30年代初，胡仁钦和科尔莫戈罗夫在20世纪30年代初建立了概率研讨会，这些研讨会由研究者参加，如[[Eugene Slutsky]]]等和[[尼古拉·斯米尔诺夫（数学家）|尼古拉·斯米尔诺夫]]，<ref name=“Vere-Jones2006page1”>{cite book | last1=Vere Jones | first1=David | title=统计科学百科全书| chapter=Khinchin，Aleksandr Yakovlevich | page=1 | year=2006 | doi=10.1002/0471667196.ess6027.pub2 | isbn=978-0471667193}}</ref>还有金钦给出了第一个随机变量的数学定义，把随机过程作为以实数线索引的一组随机变量。<ref name=“Doob1934”/><ref name=“Vere-Jones2006page4”>{cite book | last1=Vere Jones | first1=David | title=统计科学百科全书| chapter=Khinchin，Aleksandr Yakovlevich | page=4 | year=2006 | doi=10.1002/0471667196.ess6027.pub2 | isbn=978-0471667193}}</ref>{{efn | Doob在引用Khinchin时，使用了“机会变量”这个词，它曾经是“随机变量”的替代词。<ref name=“Snell2005”>{cite journal | last1=Snell | first1=J.Laurie | title=讣告：journal Leonard Doob | journal=journal of Applied Probability | volume=42 | issue=1 | year=2005 | page=251 | issn=0021-9002 | doi=10.1239/jap/1110381384 | doi access=free}</ref>}}

[[File:Joseph Doob.jpg|thumb|right|数学家[[Joseph Doob]]在随机过程理论方面做了早期的工作，做出了基本贡献，尤其是在鞅理论方面。<ref name="Getoor2009"/><ref name="Snell2005"/>他的书《随机过程》被认为在概率论领域具有很高的影响力。<ref name="Bingham2005"/>]]

In 1933 Andrei Kolmogorov published in German, his book on the foundations of probability theory titled ''Grundbegriffe der Wahrscheinlichkeitsrechnung'',{{efn|Later translated into English and published in 1950 as Foundations of the Theory of Probability<ref name="Bingham2000"/>}} where Kolmogorov used measure theory to develop an axiomatic framework for probability theory. The publication of this book is now widely considered to be the birth of modern probability theory, when the theories of probability and stochastic processes became parts of mathematics.<ref name="Bingham2000"/><ref name="Cramer1976"/>

===二战后的随机过程 Stochastic processes after World War II===

在科尔莫戈洛夫的书出版后，钦钦和科尔莫戈洛夫以及其他数学家如[[Joseph Doob]]、[[William Feller]]、[[Maurice Fréchet]]、[[Paul Lévy（数学家）| Paul Lévy]]、[[Wolfgang Doeblin]]等对概率论和随机过程进行了进一步的基础性工作，和[[Harald Cramér]]。<ref name=“Bingham2000”/><ref name=“Cramer1976”/>

同样从20世纪40年代开始，随机过程（尤其是鞅）与[[势理论]]的数学领域之间建立了联系，Shizuo Kakutani的早期思想和Joseph Doob后来的工作。<ref name="Meyer2009"/>在1950年代[[Gilbert Hunt]]完成了被认为是开创性的进一步工作，把马尔可夫过程和势理论联系起来，这对Lévy过程理论产生了重大影响，并使人们对用Itô开发的方法研究马尔可夫过程产生了更多的兴趣<ref name="JarrowProtter2004"/><ref name="Bertoin1998pageVIIIandIX">{{cite book|author=Jean Bertoin|title=Lévy Processes|url=https://books.google.com/books?id=ftcsQgMp5cUC&pg=PR8|year=1998|publisher=Cambridge University Press|isbn=978-0-521-64632-1|page=viii and ix}}</ref><ref name="Steele2012page176">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=176}}</ref>

几十年后，克莱姆把20世纪30年代称为“数学概率论的英雄时期”。<ref name=“Cramer1976”/>[[第二次世界大战]]极大地中断了概率论的发展，例如，Feller从[[瑞典]]迁移到[[美国]]<ref name=“Cramer1976”/>以及现在被认为是随机过程先驱的Doeblin之死，1915-1940；journal=The Annals of Probability | volume=19 | issue=3 | year=1991 | pages=929-934 | issn=0091-1798 | doi=10.1214/aop/1176990329 | doi access=free}</ref>

1953年，Joseph Doob出版了《随机过程 Stochastic processes》一书，这本书对随机过程理论产生了重大影响，并强调了测度理论在概率论中的重要性。<ref name="Meyer2009"/>

[[文件：Joseph Doob.jpg|thumb |右|数学家[[Joseph Doob]]在随机过程理论方面做了早期的工作，做出了基本贡献，尤其是在鞅理论方面。<ref name=“Getoor2009”/><ref name=“Snell2005”/>他的书《随机过程》被认为在概率论领域具有很高的影响力。<refname=“Bingham2005”/>]]

<ref name="Bingham2005">{{cite journal|last1=Bingham|first1=N. H.|title=Doob: a half-century on|journal=Journal of Applied Probability|volume=42|issue=1|year=2005|pages=257–266|issn=0021-9002|doi=10.1239/jap/1110381385|doi-access=free}}</ref>Doob还主要发展了鞅理论，后来Paul-André Meyer也作出了重大贡献。早期的研究是由Sergei Bernstein、Paul Lévy和Jean Ville进行的，后者采用了随机过程的鞅项。<ref name="HallHeyde2014page1">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year=2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|pages=1, 2}}</ref><ref name="Dynkin1989">{{cite journal|last1=Dynkin|first1=E. B.|title=Kolmogorov and the Theory of Markov Processes|journal=The Annals of Probability|volume=17|issue=3|year=1989|pages=822–832|issn=0091-1798|doi=10.1214/aop/1176991248|doi-access=free}}</ref>鞅理论中的方法已成为解决各种概率问题的常用方法。研究马尔可夫过程的技术和理论发展到鞅上。相反地，从鞅理论中也建立了处理Markov过程的方法。<ref name="Meyer2009"/>

After World War II the study of probability theory and stochastic processes gained more attention from mathematicians, with significant contributions made in many areas of probability and mathematics as well as the creation of new areas.<ref name="Cramer1976"/><ref name="Meyer2009">{{cite journal|last1=Meyer|first1=Paul-André|title=Stochastic Processes from 1950 to the Present|journal=Electronic Journal for History of Probability and Statistics|volume=5|issue=1|year=2009|pages=1–42}}</ref> Starting in the 1940s, [[Kiyosi Itô]] published papers developing the field of [[stochastic calculus]], which involves stochastic [[integrals]] and stochastic [[differential equations]] based on the Wiener or Brownian motion process.<ref name="Ito1998Prize">{{cite journal|title=Kiyosi Itô receives Kyoto Prize|journal=Notices of the AMS|volume=45|issue=8|year=1998|pages=981–982}}</ref>

概率的其他领域也被发展和用于研究随机过程，其中一个主要方法是大偏差理论。<ref name="Meyer2009"/> 该理论在统计物理等领域有许多应用，其核心思想至少可以追溯到20世纪30年代。20世纪60年代和70年代后期，苏联的亚历山大·温策尔和美利坚合众国的[[Monroe D.Donsker]]和[[Srinivasa Varadhan]]完成了基础工作，<ref name="Ellis1995page98">{{cite journal|last1=Ellis|first1=Richard S.|title=An overview of the theory of large deviations and applications to statistical mechanics|journal=Scandinavian Actuarial Journal|volume=1995|issue=1|year=1995|page=98|issn=0346-1238|doi=10.1080/03461238.1995.10413952}}</ref>这将使瓦拉丹获得2007年阿贝尔奖。<ref name="RaussenSkau2008">{{cite journal|last1=Raussen|first1=Martin|last2=Skau|first2=Christian|title=Interview with Srinivasa Varadhan|journal=Notices of the AMS|volume=55|issue=2|year=2008|pages=238–246}}</ref>上世纪90年代和2000年代的理论[[施拉姆–Loewner演化]]]<ref name="HenkelKarevski2012page113">{{cite book|author1=Malte Henkel|author2=Dragi Karevski|title=Conformal Invariance: an Introduction to Loops, Interfaces and Stochastic Loewner Evolution|url=https://books.google.com/books?id=fnCQWd0GEZ8C&pg=PA113|year=2012|publisher=Springer Science & Business Media|isbn=978-3-642-27933-1|page=113}}</ref>和[[粗略路径]]<ref name="FrizVictoir2010page571">{{cite book|author1=Peter K. Friz|author2=Nicolas B. Victoir|title=Multidimensional Stochastic Processes as Rough Paths: Theory and Applications|url=https://books.google.com/books?id=CVgwLatxfGsC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48721-4|page=571}}</ref>被引入和发展来研究概率论中的随机过程和其他数学对象，分别在2008年和2014年分别授予Wendelin Werner<ref name="Werner2004Fields">{{cite journal|title=2006 Fields Medals Awarded|journal=Notices of the AMS|volume=53|issue=9|year=2015|pages=1041–1044}}</ref>和Martin Hairer菲尔兹奖。<ref name="Hairer2004Fields">{{cite journal|last1=Quastel|first1=Jeremy|title=The Work of the 2014 Fields Medalists|journal=Notices of the AMS|volume=62|issue=11|year=2015|pages=1341–1344}}</ref>

Also starting in the 1940s, connections were made between stochastic processes, particularly martingales, and the mathematical field of [[potential theory]], with early ideas by [[Shizuo Kakutani]] and then later work by Joseph Doob.<ref name="Meyer2009"/> Further work, considered pioneering, was done by [[Gilbert Hunt]] in the 1950s, connecting Markov processes and potential theory, which had a significant effect on the theory of Lévy processes and led to more interest in studying Markov processes with methods developed by Itô.<ref name="JarrowProtter2004"/><ref name="Bertoin1998pageVIIIandIX">{{cite book|author=Jean Bertoin|title=Lévy Processes|url=https://books.google.com/books?id=ftcsQgMp5cUC&pg=PR8|year=1998|publisher=Cambridge University Press|isbn=978-0-521-64632-1|page=viii and ix}}</ref><ref name="Steele2012page176">{{cite book|author=J. Michael Steele|title=Stochastic Calculus and Financial Applications|url=https://books.google.com/books?id=fsgkBAAAQBAJ&pg=PR4|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4684-9305-4|page=176}}</ref>

随机过程理论仍然是研究的焦点，每年都有关于随机过程的国际会议。<ref name="BlathImkeller2011"/><ref name="Applebaum2004page1336"/>

In 1953 Doob published his book ''Stochastic processes'', which had a strong influence on the theory of stochastic processes and stressed the importance of measure theory in probability.<ref name="Meyer2009"/>

<references/>

<ref name="Bingham2005">{{cite journal|last1=Bingham|first1=N. H.|title=Doob: a half-century on|journal=Journal of Applied Probability|volume=42|issue=1|year=2005|pages=257–266|issn=0021-9002|doi=10.1239/jap/1110381385|doi-access=free}}</ref> Doob also chiefly developed the theory of martingales, with later substantial contributions by [[Paul-André Meyer]]. Earlier work had been carried out by [[Sergei Bernstein]], [[Paul Lévy (mathematician)|Paul Lévy]] and [[Jean Ville]], the latter adopting the term martingale for the stochastic process.<ref name="HallHeyde2014page1">{{cite book|author1=P. Hall|author2=C. C. Heyde|title=Martingale Limit Theory and Its Application|url=https://books.google.com/books?id=gqriBQAAQBAJ&pg=PR10|year=2014|publisher=Elsevier Science|isbn=978-1-4832-6322-9|pages=1, 2}}</ref><ref name="Dynkin1989">{{cite journal|last1=Dynkin|first1=E. B.|title=Kolmogorov and the Theory of Markov Processes|journal=The Annals of Probability|volume=17|issue=3|year=1989|pages=822–832|issn=0091-1798|doi=10.1214/aop/1176991248|doi-access=free}}</ref> Methods from the theory of martingales became popular for solving various probability problems. Techniques and theory were developed to study Markov processes and then applied to martingales. Conversely, methods from the theory of martingales were established to treat Markov processes.<ref name="Meyer2009"/>

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概率的其他领域也被发展和用于研究随机过程，其中一个主要方法是大偏差理论。<ref name=“Meyer2009”/>该理论在统计物理等领域有许多应用，其核心思想至少可以追溯到20世纪30年代。20世纪60年代和70年代后期，苏联的亚历山大·温策尔和美利坚合众国的[[Monroe D.Donsker]]和[[Srinivasa Varadhan]]完成了基础工作，<ref name=“Ellis1995page98”>{cite journal | last1=Ellis | first1=Richard S.| title=大理论概述统计力学的偏差与应用| journal=斯堪的纳维亚精算杂志| volume=1995 | issue=1 | year=1995 | page=98 | issn=0346-1238 | doi=10.1080/03461238.1995.10413952}</ref>，这将使瓦拉丹获得2007年阿贝尔奖。<ref name=“RaussenSkau2008”>{cite journal | last1=Raussen | first1=Martin | last2=Skau | title=采访Srinivasa Varadhan | journal=AMS通知|卷=55 |问题=2 |年=2008 |页=238–246}</ref>上世纪90年代和2000年代的理论[[施拉姆–Loewner演化]]]<ref name=“HenkelkeKarevskI2012Page113”>{引用书〈引书| AuthorAuthorAuthorAuthorAuthorAuthorAuthorAuth1=马尔特-汉克尔| author2=德拉吉Karevski | title=共形不变性：循环、接口和随机Loewner演化简介| url=https://books.google.com/books？id=fnCQWd0GEZ8C&pg=PA113 | year=2012 | publisher=Springer Science&Business Media | isbn=978-3-642-27933-1 | page=113}</ref>和[[粗略路径]]<ref name=“frizvictoir201page571”>{cite book | author1=Peter K.Friz | author2=Nicolas B.Victoir | title=多维随机过程作为粗糙路径：理论和应用程序| url=https://books.google.com/books？id=CVgwLatxfGsC | year=2010 | publisher=Cambridge University Press | isbn=978-1-139-48721-4 | page=571}</ref>被引入和发展来研究概率论中的随机过程和其他数学对象，分别在2008年和2014年分别授予[[Wendelin Werner]]<ref name=“Werner2004Fields”>{cite journal | title=2006菲尔兹勋章| journal=AMS通知|卷=53 |问题=9 |年=2015 |页=1041-1044}</ref>和2014年授予[[Martin Haier]]journal | last1=Quastel | first1=Jeremy | title=2014年菲尔兹奖获得者的作品| journal=AMS的通知|卷=62 |问题=11 |年=2015 |页=1341-1344}</ref>

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[[Category:随机模型]]

[[Category:随机模型]]

[[Category:统计数据类型]]

[[Category:统计数据类型]]