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When interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers, then the stochastic process is said to be in discrete time. If the index set is some interval of the real line, then time is said to be continuous. The two types of stochastic processes are respectively referred to as discrete-time and continuous-time stochastic processes. Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge, particularly due to the index set being uncountable. If the index set is the integers, or some subset of them, then the stochastic process can also be called a random sequence. 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". This phrase was used, with reference to Bernoulli, by Ladislaus Bortkiewicz 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. though the German term had been used earlier, for example, by Andrei Kolmogorov in 1931.

When interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers, then the stochastic process is said to be in discrete time. If the index set is some interval of the real line, then time is said to be continuous. The two types of stochastic processes are respectively referred to as discrete-time and continuous-time stochastic processes. Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge, particularly due to the index set being uncountable. If the index set is the integers, or some subset of them, then the stochastic process can also be called a random sequence. 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". This phrase was used, with reference to Bernoulli, by Ladislaus Bortkiewicz 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. though the German term had been used earlier, for example, by Andrei Kolmogorov in 1931.

当被解释为时间时，如果一个随机过程的索引集的元素数量有限或可数，如一个有限的数字集，一个整数集，或自然数集，那么随机过程被称为在离散时间内随机。如果索引集是实线的某个区间，那么时间就是连续的。这两类随机过程分别称为离散时间过程和连续时间过程。离散时间随机过程被认为更容易研究，因为连续时间过程需要更先进的数学技术和知识，特别是由于索引集是不可数的。如果索引集是整数，或者整数的一些子集，那么随机过程也可以被称为'''<font color="#ff8000"> 随机序列Random sequence</font>'''。雅各布 · 伯努利在1713年以拉丁文出版的《猜测概率论》一书中使用了“猜测随机论”这个短语，这个短语被翻译成了“猜测或推测的艺术”。1917年，拉迪斯劳斯·博特基威茨在德语中写下了“随机”一词，意思是随机。1934年，Joseph Doob 在一篇论文中首次提到随机过程这个词。尽管这个德语术语早在1931年就被安德烈 · 科尔莫哥罗夫使用过。

当解释为时间时，如果随机过程的索引集有有限个或可数的元素，例如有限的一组数字、一组整数或自然数，则该随机过程称为离散时间的。如果索引集是实数轴的某个区间，则时间被称为连续的。这两类随机过程分别称为<font color="#ff8000"> 离散时间随机过程</font>和<font color="#ff8000"> 连续时间随机过程</font>。离散时间随机过程被认为更容易研究，因为连续时间过程需要更先进的数学技术和知识，特别当索引集不可数时。如果索引集是整数或整数的子集，那么随机过程也可以称为<font color="#ff8000"> 随机序列</font>。Jakob Bernoulli于1713年在拉丁文中首次发表了关于概率的著作Ars Conspectandi，他使用了“Ars consuctandi-sive-Stochastice”一词，该词已被翻译为“推测或随机的艺术”。这个短语是由Ladislaus Bortkiewicz在1917年用德语写下的单词stochastik，意思是随机的。“随机过程”一词最早出现在1934年约瑟夫·杜布的一篇论文中。尽管德语这个词在早些时候就被使用过，例如，安德烈·科尔莫戈洛夫（Andrei Kolmogorov）在1931年就使用过。

 

--[[用户:水流心不竞|水流心不竞]]（[[用户讨论:水流心不竞|讨论]]）如果一个随机过程的索引集的元素数量有限或可数，如一个有限的数字集，一个整数集，或自然数集，那么随机过程被称为在离散时间内随机。如果索引集是实线的某个区间，那么时间就是连续的  修改润色语言  “实线的某个区间”→在实数轴上的某个区间上

A stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set.<ref name="Parzen1999"/><ref name="GikhmanSkorokhod1969page1"/> The set used to index the random variables is called the '''index set'''. Historically, the index set was some [[subset]] of the [[real line]], such as the [[natural numbers]], giving the index set the interpretation of time.<ref name="doob1953stochasticP46to47"/> Each random variable in the collection takes values from the same [[mathematical space]] known as the '''state space'''. This state space can be, for example, the integers, the real line or <math>n</math>-dimensional Euclidean space.<ref name="doob1953stochasticP46to47"/><ref name="GikhmanSkorokhod1969page1"/> An '''increment''' is the amount that a stochastic process changes between two index values, often interpreted as two points in time.<ref name="KarlinTaylor2012page27"/><ref name="Applebaum2004page1337"/> A stochastic process can have many [[Outcome (probability)|outcomes]], due to its randomness, and a single outcome of a stochastic process is called, among other names, a '''sample function''' or '''realization'''.<ref name="Lamperti1977page1"/><ref name="RogersWilliams2000page121b"/>

A stochastic or random process can be defined as a collection of random variables that is indexed by some mathematical set, meaning that each random variable of the stochastic process is uniquely associated with an element in the set.<ref name="Parzen1999"/><ref name="GikhmanSkorokhod1969page1"/> The set used to index the random variables is called the '''index set'''. Historically, the index set was some [[subset]] of the [[real line]], such as the [[natural numbers]], giving the index set the interpretation of time.<ref name="doob1953stochasticP46to47"/> Each random variable in the collection takes values from the same [[mathematical space]] known as the '''state space'''. This state space can be, for example, the integers, the real line or <math>n</math>-dimensional Euclidean space.<ref name="doob1953stochasticP46to47"/><ref name="GikhmanSkorokhod1969page1"/> An '''increment''' is the amount that a stochastic process changes between two index values, often interpreted as two points in time.<ref name="KarlinTaylor2012page27"/><ref name="Applebaum2004page1337"/> A stochastic process can have many [[Outcome (probability)|outcomes]], due to its randomness, and a single outcome of a stochastic process is called, among other names, a '''sample function''' or '''realization'''.<ref name="Lamperti1977page1"/><ref name="RogersWilliams2000page121b"/>

[[File:Wiener process 3d.png|thumb|right|A single computer-simulated '''sample function''' or '''realization''', among other terms, of a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2. The index set of this stochastic process is the non-negative numbers, while its state space is three-dimensional Euclidean space.]]

[[File:Wiener process 3d.png|thumb|right|A single computer-simulated '''sample function''' or '''realization''', among other terms, of a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2. The index set of this stochastic process is the non-negative numbers, while its state space is three-dimensional Euclidean space.]]

[[文件：维纳工艺3d.png | thumb | right |单个计算机模拟时间0≤t≤2的三维Wiener或Brownian运动过程的“样本函数”或“实现”。这个随机过程的指标集是非负数，而其状态空间是三维欧几里德空间

[[文件：维纳工艺3d.png | thumb | right |单个计算机模拟时间0≤t≤2的三维Wiener或Brownian运动过程的“样本函数”或“实现”。这个随机过程的指标集是非负数，而其状态空间是三维欧几里德空间]]

===Classifications分类===

===Classifications分类===

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.

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.

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。

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（stochastic）过程和随机（random）过程，当索引集是 <math>n</math>-维欧几里得空间 <math>\mathbb{R}^n</math>或者是流形时，则使用随机场。虽然<math>\{X(t)\}</math>被认为是对函数表示法的滥用，但<math>\{X(t)\}</math>还是被简单地称为<math>X</math> 或 <math>X(t)</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>

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>

随机过程可以用不同的方法进行分类，例如，根据其状态空间、指标集或随机变量之间的相关性。一种常见的分类方法是通过索引集和状态空间的[[基数]]=https://books.google.com/books？id=dSDxjX9nmmMC | year=2012 | publisher=academical Press | isbn=978-0-08-057041-9 | page=26}</ref>{cite book | author1=Donald L.Snyder | author2=Michael I.Miller | title=时空中的随机点过程| url=https://books.google.com/books？id=c|3UBwAAQBAJ|year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4612-3166-0 |页面=24，25}</ref>

随机过程可以用不同的方法进行分类，例如，根据其状态空间、索引集或随机变量之间的相关性。一种常见的分类方法是通过索引集和状态空间的[[基数]]进行分类。<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>

When interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers, then the stochastic process is said to be in '''[[discrete time]]'''.<ref name="Billingsley2008page482"/><ref name="Borovkov2013page527">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=527}}</ref> If the index set is some interval of the real line, then time is said to be '''[[continuous time|continuous]]'''. The two types of stochastic processes are respectively referred to as '''discrete-time''' and '''[[continuous-time stochastic process]]es'''.<ref name="KarlinTaylor2012page27"/><ref name="Brémaud2014page120"/><ref name="Rosenthal2006page177">{{cite book|author=Jeffrey S Rosenthal|title=A First Look at Rigorous Probability Theory|url=https://books.google.com/books?id=am1IDQAAQBAJ|year=2006|publisher=World Scientific Publishing Co Inc|isbn=978-981-310-165-4|pages=177–178}}</ref> Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge, particularly due to the index set being uncountable.<ref name="KloedenPlaten2013page63">{{cite book|author1=Peter E. Kloeden|author2=Eckhard Platen|title=Numerical Solution of Stochastic Differential Equations|url=https://books.google.com/books?id=r9r6CAAAQBAJ=PA1|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-12616-5|page=63}}</ref><ref name="Khoshnevisan2006page153">{{cite book|author=Davar Khoshnevisan|title=Multiparameter Processes: An Introduction to Random Fields|url=https://books.google.com/books?id=XADpBwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21631-7|pages=153–155}}</ref> If the index set is the integers, or some subset of them, then the stochastic process can also be called a '''random sequence'''.<ref name="Borovkov2013page527"/>

When interpreted as time, if the index set of a stochastic process has a finite or countable number of elements, such as a finite set of numbers, the set of integers, or the natural numbers, then the stochastic process is said to be in '''[[discrete time]]'''.<ref name="Billingsley2008page482"/><ref name="Borovkov2013page527">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=527}}</ref> If the index set is some interval of the real line, then time is said to be '''[[continuous time|continuous]]'''. The two types of stochastic processes are respectively referred to as '''discrete-time''' and '''[[continuous-time stochastic process]]es'''.<ref name="KarlinTaylor2012page27"/><ref name="Brémaud2014page120"/><ref name="Rosenthal2006page177">{{cite book|author=Jeffrey S Rosenthal|title=A First Look at Rigorous Probability Theory|url=https://books.google.com/books?id=am1IDQAAQBAJ|year=2006|publisher=World Scientific Publishing Co Inc|isbn=978-981-310-165-4|pages=177–178}}</ref> Discrete-time stochastic processes are considered easier to study because continuous-time processes require more advanced mathematical techniques and knowledge, particularly due to the index set being uncountable.<ref name="KloedenPlaten2013page63">{{cite book|author1=Peter E. Kloeden|author2=Eckhard Platen|title=Numerical Solution of Stochastic Differential Equations|url=https://books.google.com/books?id=r9r6CAAAQBAJ=PA1|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-12616-5|page=63}}</ref><ref name="Khoshnevisan2006page153">{{cite book|author=Davar Khoshnevisan|title=Multiparameter Processes: An Introduction to Random Fields|url=https://books.google.com/books?id=XADpBwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21631-7|pages=153–155}}</ref> If the index set is the integers, or some subset of them, then the stochastic process can also be called a '''random sequence'''.<ref name="Borovkov2013page527"/>

当解释为时间时，如果随机过程的指标集有有限个或可数个元素，例如有限的一组数、一组整数或自然数，那么随机过程被称为“'[[离散时间]]''”。<ref name=“Billingsley2008page482”/><ref name=“Borovkov2013page527”>{cite book | author=Alexander A.Borovkov | title=Probability Theory |网址=https://books.google.com/books？id=hRk_AAAAQBAJ | year=2013 | publisher=Springer Science&Business Media | isbn=978-1-4471-5201-9 | page=527}}</ref>如果索引集是实线的某个区间，则时间被称为“'[[continuous time|continuous]]”。这两类随机过程分别被称为“离散时间”和“[[连续时间随机过程]]es”。<ref name=“KarlinTaylor2012page27”/><ref name=“Brémaud2014page120”/><ref name=“Rosenthal2006page177”>{cite book | author=Jeffrey S Rosenthal | title=A First Look on critical Probability理论|网址=https://books.google.com/books？id=am1IDQAAQBAJ | year=2006 | publisher=World Scientific Publishing Co Inc | isbn=978-981-310-165-4 | pages=177-178}</ref>离散时间随机过程被认为更容易研究，因为连续时间过程需要更先进的数学技术和知识，特别是由于索引集是不可数的。<ref name=“KloedenPlaten2013page63”>{cite book | author1=Peter E.Kloeden | author2=Eckhard Platen | title=随机微分方程的数值解=https://books.google.com/books？id=r9r6CAAAQBAJ=PA1 | year=2013 | publisher=Springer Science&Business Media | isbn=978-3-662-12616-5 | page=63}</ref><ref name=“khoshnivesan2006page153”>{cite book | author=Davar khoshnivesan | title=多参数过程：随机字段简介| url=https://books.google.com/books？id=XADpBwAAQBAJ | year=2006 | publisher=Springer Science&Business Media | isbn=978-0-387-21631-7 | pages=153–155}</ref>如果索引集是整数或整数的子集，则随机过程也可以称为“随机序列”。<ref name=“Borovkov2013page527”/>

当解释为时间时，如果随机过程的指标集有有限个或可数个元素，例如有限的一组数、一组整数或自然数，那么随机过程被称为“[[离散时间]]”<ref name="Billingsley2008page482"/><ref name="Borovkov2013page527">{{cite book|author=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=527}}</ref> 。如果索引集是实数轴上的某个区间，则时间被称为“'[[连续时间]]”。这两类随机过程分别被称为“离散时间”和“[[连续时间随机过程]]es”<ref name="KarlinTaylor2012page27"/><ref name="Brémaud2014page120"/><ref name="Rosenthal2006page177">{{cite book|author=Jeffrey S Rosenthal|title=A First Look at Rigorous Probability Theory|url=https://books.google.com/books?id=am1IDQAAQBAJ|year=2006|publisher=World Scientific Publishing Co Inc|isbn=978-981-310-165-4|pages=177–178}}</ref>。离散时间随机过程被认为更容易研究，因为连续时间过程需要更先进的数学技术和知识，特别当索引集不可数时。<ref name="KloedenPlaten2013page63">{{cite book|author1=Peter E. Kloeden|author2=Eckhard Platen|title=Numerical Solution of Stochastic Differential Equations|url=https://books.google.com/books?id=r9r6CAAAQBAJ=PA1|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-12616-5|page=63}}</ref><ref name="Khoshnevisan2006page153">{{cite book|author=Davar Khoshnevisan|title=Multiparameter Processes: An Introduction to Random Fields|url=https://books.google.com/books?id=XADpBwAAQBAJ|year=2006|publisher=Springer Science & Business Media|isbn=978-0-387-21631-7|pages=153–155}}</ref> 如果索引集是整数或整数的子集，则随机过程也可以称为“随机序列”。<ref name=“Borovkov2013page527”/>

If the state space is the integers or natural numbers, then the stochastic process is called a '''discrete''' or '''integer-valued stochastic process'''. If the state space is the real line, then the stochastic process is referred to as a '''real-valued stochastic process''' or a '''process with continuous state space'''. If the state space is <math>n</math>-dimensional Euclidean space, then the stochastic process is called a <math>n</math>-'''dimensional vector process''' or <math>n</math>-'''vector process'''.<ref name="Florescu2014page294"/><ref name="KarlinTaylor2012page26"/>

If the state space is the integers or natural numbers, then the stochastic process is called a '''discrete''' or '''integer-valued stochastic process'''. If the state space is the real line, then the stochastic process is referred to as a '''real-valued stochastic process''' or a '''process with continuous state space'''. If the state space is <math>n</math>-dimensional Euclidean space, then the stochastic process is called a <math>n</math>-'''dimensional vector process''' or <math>n</math>-'''vector process'''.<ref name="Florescu2014page294"/><ref name="KarlinTaylor2012page26"/>

如果状态空间是整数或自然数，则随机过程称为“离散”或“整值随机过程”。如果状态空间是实线，则随机过程被称为“实值随机过程”或“具有连续状态空间的过程”。如果状态空间是<math>n</math>-维欧几里德空间，则随机过程称为<math>n</math>-“维向量过程”或<math>n</math>—“向量过程”。<ref name=“florescu214page294”/><ref name=“KarlinTaylor2012page26”/>

如果状态空间是整数或自然数，则随机过程称为“离散”或“整值随机过程”。如果状态空间是实数轴，则随机过程被称为“实值随机过程”或“具有连续状态空间的过程”。如果状态空间是<math>n</math>-维欧几里德空间，则随机过程称为<math>n</math>-“维向量过程”或<math>n</math>—“向量过程”。<ref name=“florescu214page294”/><ref name=“KarlinTaylor2012page26”/>

Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. But some also use the term to refer to processes that change in continuous time, particularly the Wiener process used in finance, which has led to some confusion, resulting in its criticism. There are other various types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines.

Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. But some also use the term to refer to processes that change in continuous time, particularly the Wiener process used in finance, which has led to some confusion, resulting in its criticism. There are other various types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines.

'''<font color="#ff8000"> 随机游走Random walks</font>'''是随机过程，通常定义为欧氏空间中的等价随机变量或随机向量的和，因此它们是在离散时间中变化的过程。但有些人也用这个词来指连续时间中发生变化的过程，特别是在金融领域使用的维纳过程，这种过程导致了一些混淆，从而招致了批评。还有其他各种类型的'''<font color="#ff8000"> 随机游走Random walks</font>'''，定义它们的状态空间可以是其他数学对象，如格子和群，一般来说，它们被高度研究，在不同学科中有许多应用。

'''<font color="#ff8000"> 随机游走Random walks</font>'''是一种随机过程，通常定义为欧氏空间中的等价随机变量或随机向量的和，因此它们是在离散时间中变化的过程。但有些人也用这个词来指连续时间中发生变化的过程，特别是在金融领域使用的维纳过程，这种过程导致了一些混淆，因此招致了批评。还有其他各种类型的'''<font color="#ff8000"> 随机游走</font>'''，定义它们的状态空间可以是其他数学对象，如格子和群。一般来说，它们被高度研究，在不同学科中有许多应用。

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.

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.

一个经典的'''<font color="#ff8000"> 随机游走Random walk</font>'''的例子被称为'''<font color="#ff8000"> 简单随机游走SimpleRandom walk</font>'''，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'''<font color="#ff8000">伯努利过程Bernoulli process</font>'''，其中每个 伯努利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<math>p</math>的增加而增加1，或随概率<math>1-p</math>的减少而减少1，所以这种'''<font color="#ff8000"> 随机游走Random walk</font>'''的索引集是自然数，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'''<font color="#ff8000"> 对称随机游走Symmetric Random walk</font>'''。

一个经典的'''<font color="#ff8000"> 随机游走</font>'''的例子被称为'''<font color="#ff8000"> 简单随机游走SimpleRandom walk</font>'''，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'''<font color="#ff8000">伯努利过程Bernoulli process</font>'''，其中每个 伯努利Bernoulli 变量要么取值为正，要么取值为负。换句话说，简单随机游走发生在整数上，它的值随概率<math>p</math>的增加而增加1，或随概率<math>1-p</math>的减少而减少1，所以这种'''<font color="#ff8000"> 随机游走</font>'''的索引集是自然数，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'''<font color="#ff8000"> 对称随机游走Symmetric Random walk</font>'''。

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>

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>

在[[英语语言|英语]]中，“随机”一词最初用作形容词，其定义是“与推测有关”，源于一个[[希腊语|希腊语]]一词，意思是“瞄准一个标记，猜测”，而[[牛津英语词典]]将1662年作为最早出现的年份。<ref name=“Oxfordstraphic”>{Cite OED | random}</ref>在他关于概率“Ars conquectandi”的著作中，最初于1713年以拉丁文出版，[[Jakob Bernoulli]]使用了“Ars conquectandi istice”这个短语，这本书已经被翻译成“猜想或随机的艺术”。<ref name=“Sheĭnin2006page5”>{cite book | author=O.B.Sheĭnin|title=概率论和统计学，以简短的口述为例=https://books.google.com/books？年=2006年；publisher=NG Verlag Verlag | isbn=978-3-938417-40-9 | page=5}</ref>这一短语是[[Ladislaus bortkiewiccz]]]<ref name=“sheyninstrrecker2011page136”>{{引用本书〈author1=OscarSheynin；author1=奥斯卡Sheynin；author2=Heinrich Strecker；title=Alexandr A.Chuprov A.Chprov:生活，工作，工作，生活，工作，工作，生活，工作，工作，工作，生活，工作，工作，工作，工作，工作，工作，生活，工作，工作，工作，工作在，通信地址=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>他在1917年用德语写下了“随机”一词。术语“随机过程”最早出现在1934年[[Joseph Doob]]的一篇论文中。<ref name=“oxfordstractical”/>对于这个术语和一个具体的数学定义，Doob引用了另一篇1934年的论文，其中[[Aleksandr Khinchin]]在德语中使用了术语“随机过程”，<ref name=“Doob1934”/><<ref name=“Doob1934”>>{{引用期刊| last1=Khintchine | first1=A.| title=KorrationationTheOrie Oreider StatinerenStochastische陈Prozessese | journal=MatheMatheMatche Annalen；volume=109；volume=109；问题=1年=1934年|年=1934 |第604–615 | Issnn=00225-5831 | doi=10.1007/BF014449156}}}}}{}}}}例如，尽管德语这个词在早些时候被使用过，安德烈科莫戈罗夫于1931年由Andrei Kolmogorov1931年由Andrei Kolmogorov1931年的Andrei Kolmogorov1931Page1>{〈引用期刊| last1=Kolmogoroff 124；first1=A.| title=ÜBerDie Analyt陈Methoden在德Wahrscheinlickeitsrcherechnung | journal=MatheMatheMatheAnnanLen；卷=104 |问题=1年=1931年|页=1 |页=1 | issn=00225-5831 | 124; doi=10.1007/}}</ref>

在[[英语语言|英语]]中，“随机”一词最初用作形容词，其定义是“与推测有关”，源于一个[[希腊语|希腊语]]词，意思是“瞄准一个标记，猜测”，而[[牛津英语词典]]将1662年作为最早出现的年份。<ref name=“Oxfordstraphic”>{Cite OED | random}</ref>在他关于概率“Ars conquectandi”的著作中，最初于1713年以拉丁文出版，[[Jakob Bernoulli]]使用了“Ars conquectandi istice”这个短语，这本书已经被翻译成“猜想或随机的艺术”。<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>这一短语是[[拉迪斯劳斯·博特基维茨]]]在关于伯努利问题中使用，<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>他在1917年用德语写下了“随机”一词。术语“随机过程”最早出现在1934年[[Joseph Doob]]的一篇论文中。<ref name=“oxfordstractical”/>对于这个术语和一个具体的数学定义，Doob引用了另一篇1934年的论文，其中[[Aleksandr Khinchin]]在德语中使用了术语“随机过程”，<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>尽管德语这个词在早些时候就被使用过，例如，安德烈·科尔莫戈洛夫（Andrei Kolmogorov）在1931年就使用过。<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>

The definition of a stochastic process varies,<ref name="FristedtGray2013page580">{{cite book|author1=Bert E. Fristedt|author2=Lawrence F. Gray|title=A Modern Approach to Probability Theory|url=https://books.google.com/books?id=9xT3BwAAQBAJ&pg=PA716|year= 2013|publisher=Springer Science & Business Media|isbn=978-1-4899-2837-5|page=580}}</ref> but a stochastic process is traditionally defined as a collection of random variables indexed by some set.<ref name="RogersWilliams2000page121"/><ref name="Asmussen2003page408"/> The terms ''random process'' and ''stochastic process'' are considered synonyms and are used interchangeably, without the index set being precisely specified.<ref name="Kallenberg2002page24"/><ref name="ChaumontYor2012"/><ref name="AdlerTaylor2009page7"/><ref name="Stirzaker2005page45">{{cite book|author=David Stirzaker|title=Stochastic Processes and Models|url=https://books.google.com/books?id=0avUelS7e7cC|year=2005|publisher=Oxford University Press|isbn=978-0-19-856814-8|page=45}}</ref><ref name="Rosenblatt1962page91">{{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/91 91]}}</ref><ref name="Gubner2006page383">{{cite book|author=John A. Gubner|title=Probability and Random Processes for Electrical and Computer Engineers|url=https://books.google.com/books?id=pa20eZJe4LIC|year=2006|publisher=Cambridge University Press|isbn=978-1-139-45717-0|page=383}}</ref> Both "collection",<ref name="Lamperti1977page1"/><ref name="Stirzaker2005page45"/> or "family" are used<ref name="Parzen1999"/><ref name="Ito2006page13">{{cite book|author=Kiyosi Itō|title=Essentials of Stochastic Processes|url=https://books.google.com/books?id=pY5_DkvI-CcC&pg=PR4|year=2006|publisher=American Mathematical Soc.|isbn=978-0-8218-3898-3|page=13}}</ref> while instead of "index set", sometimes the terms "parameter set"<ref name="Lamperti1977page1"/> or "parameter space"<ref name="AdlerTaylor2009page7"/> are used.

The definition of a stochastic process varies,<ref name="FristedtGray2013page580">{{cite book|author1=Bert E. Fristedt|author2=Lawrence F. Gray|title=A Modern Approach to Probability Theory|url=https://books.google.com/books?id=9xT3BwAAQBAJ&pg=PA716|year= 2013|publisher=Springer Science & Business Media|isbn=978-1-4899-2837-5|page=580}}</ref> but a stochastic process is traditionally defined as a collection of random variables indexed by some set.<ref name="RogersWilliams2000page121"/><ref name="Asmussen2003page408"/> The terms ''random process'' and ''stochastic process'' are considered synonyms and are used interchangeably, without the index set being precisely specified.<ref name="Kallenberg2002page24"/><ref name="ChaumontYor2012"/><ref name="AdlerTaylor2009page7"/><ref name="Stirzaker2005page45">{{cite book|author=David Stirzaker|title=Stochastic Processes and Models|url=https://books.google.com/books?id=0avUelS7e7cC|year=2005|publisher=Oxford University Press|isbn=978-0-19-856814-8|page=45}}</ref><ref name="Rosenblatt1962page91">{{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/91 91]}}</ref><ref name="Gubner2006page383">{{cite book|author=John A. Gubner|title=Probability and Random Processes for Electrical and Computer Engineers|url=https://books.google.com/books?id=pa20eZJe4LIC|year=2006|publisher=Cambridge University Press|isbn=978-1-139-45717-0|page=383}}</ref> Both "collection",<ref name="Lamperti1977page1"/><ref name="Stirzaker2005page45"/> or "family" are used<ref name="Parzen1999"/><ref name="Ito2006page13">{{cite book|author=Kiyosi Itō|title=Essentials of Stochastic Processes|url=https://books.google.com/books?id=pY5_DkvI-CcC&pg=PR4|year=2006|publisher=American Mathematical Soc.|isbn=978-0-8218-3898-3|page=13}}</ref> while instead of "index set", sometimes the terms "parameter set"<ref name="Lamperti1977page1"/> or "parameter space"<ref name="AdlerTaylor2009page7"/> are used.

随机过程的定义是不同的，<ref name=“FristedGray2013Page580”>{cite book | author1=Bert E.Fristedt|author2=Lawrence F.Gray | title=a Modern Approach to Probability Theory |网址=https://books.google.com/books？id=9xT3BwAAQBAJ&pg=PA716 | year=2013 | publisher=Springer Science&Business Media | isbn=978-1-4899-2837-5 | page=580}}</ref>但是随机过程传统上被定义为由一些集合索引的随机变量的集合进程“”被视为同义词，可以互换使用，而无需精确指定索引集。<ref name=“KallenbergG2002Page24”/><ref name=“ChaumontYor2012”/><ref name=“adlertaylor2009 page7”/><ref name=“Stirzaker2005page45”>{cite book | author=David Stirzaker | title=random Processes and Models | url=https://books.google.com/books？id=0avUelS7e7cC | year=2005 | publisher=Oxford University Press | isbn=978-0-19-856814-8 | page=45}</ref><ref name=“Rosenblatt1962page91”>{cite book |作者=Murray Rosenblatt | title=Random Processes | url=https://archive.org/details/randomprocess00rose\u 0|url access=注册|年份=1962 | publisher=牛津大学按|页=[https://archive.org/details/randomprocess00rose_0/page/9191]}</ref><ref name=“Gubner2006page383”>{cite book | author=John A.Gubner | title=电气和计算机工程师的概率和随机过程| url=https://books.google.com/books？id=pa20eZJe4LIC | year=2006 | publisher=Cambridge University Press | isbn=978-1-139-45717-0 | page=383}</ref>两个“收藏”，<ref name=“Lamperti1977page1”/><ref name=“Stirzaker2005page45”/>或“family”用于<ref name=“Parzen1999”/><ref name=“Ito2006page13”>{cite book | author=Kiyosi Itōtitle=Essentials of randomic Processes|url=https://books.google.com/books？id=pY5_DkvI-CcC&pg=PR4 | year=2006 | publisher=American Mathematic Soc.| isbn=978-0-8218-3898-3 | page=13}}</ref>而不是“索引集”，有时使用术语“parameter set”<ref name=“Lamperti1977page1”/>或“parameter space”<ref name=“adlertaylor2009 page7”/>。

随机过程的定义是不同的，<ref name="FristedtGray2013page580">{{cite book|author1=Bert E. Fristedt|author2=Lawrence F. Gray|title=A Modern Approach to Probability Theory|url=https://books.google.com/books?id=9xT3BwAAQBAJ&pg=PA716|year= 2013|publisher=Springer Science & Business Media|isbn=978-1-4899-2837-5|page=580}}</ref> 但是随机过程传统上被定义为一组随机变量的集合<ref name="RogersWilliams2000page121"/><ref name="Asmussen2003page408"/>。术语“随机random过程”和“随机stochastic过程”被视为同义词，可以互换使用，而无需精确指定索引集。<ref name="Kallenberg2002page24"/><ref name="ChaumontYor2012"/><ref name="AdlerTaylor2009page7"/><ref name="Stirzaker2005page45">{{cite book|author=David Stirzaker|title=Stochastic Processes and Models|url=https://books.google.com/books?id=0avUelS7e7cC|year=2005|publisher=Oxford University Press|isbn=978-0-19-856814-8|page=45}}</ref><ref name="Rosenblatt1962page91">{{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/91 91]}}</ref><ref name="Gubner2006page383">{{cite book|author=John A. Gubner|title=Probability and Random Processes for Electrical and Computer Engineers|url=https://books.google.com/books?id=pa20eZJe4LIC|year=2006|publisher=Cambridge University Press|isbn=978-1-139-45717-0|page=383}}</ref>。两个“集合”<ref name="Lamperti1977page1"/><ref name="Stirzaker2005page45"/>，或“家庭”使用<ref name="Parzen1999"/><ref name="Ito2006page13">术语“参数集”<ref name="Lamperti1977page1"/> 或“参数空间”<ref name="AdlerTaylor2009page7"/> ，而不是“索引集”。

Realizations of Wiener processes (or Brownian motion processes) with drift () and without drift ().

Realizations of Wiener processes (or Brownian motion processes) with drift () and without drift ().

带漂移()和无漂移()的 '''<font color="#ff8000"> 维纳过程Wiener process</font>'''(或布朗运动过程)的实现。

带漂移()和无漂移()的 '''<font color="#ff8000"> 维纳过程</font>'''(或布朗运动过程)的实现。

The term ''random function'' is also used to refer to a stochastic or random process,<ref name="GikhmanSkorokhod1969page1"/><ref name="Loeve1978">{{cite book|author=M. Loève|title=Probability Theory II|url=https://books.google.com/books?id=1y229yBbULIC|year=1978|publisher=Springer Science & Business Media|isbn=978-0-387-90262-3|page=163}}</ref><ref name="Brémaud2014page133">{{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=133}}</ref> though sometimes it is only used when the stochastic process takes real values.<ref name="Lamperti1977page1"/><ref name="Ito2006page13"/> This term is also used when the index sets are mathematical spaces other than the real line,<ref name="GikhmanSkorokhod1969page1"/><ref name="GusakKukush2010page1">{{harvtxt|Gusak|Kukush|Kulik|Mishura|2010}}, p. 1</ref> while the terms ''stochastic process'' and ''random process'' are usually used when the index set is interpreted as time,<ref name="GikhmanSkorokhod1969page1"/><ref name="GusakKukush2010page1"/><ref name="Bass2011page1">{{cite book|author=Richard F. Bass|title=Stochastic Processes|url=https://books.google.com/books?id=Ll0T7PIkcKMC|year=2011|publisher=Cambridge University Press|isbn=978-1-139-50147-7|page=1}}</ref> 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]].<ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/><ref name="AdlerTaylor2009page7"/>

The term ''random function'' is also used to refer to a stochastic or random process,<ref name="GikhmanSkorokhod1969page1"/><ref name="Loeve1978">{{cite book|author=M. Loève|title=Probability Theory II|url=https://books.google.com/books?id=1y229yBbULIC|year=1978|publisher=Springer Science & Business Media|isbn=978-0-387-90262-3|page=163}}</ref><ref name="Brémaud2014page133">{{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=133}}</ref> though sometimes it is only used when the stochastic process takes real values.<ref name="Lamperti1977page1"/><ref name="Ito2006page13"/> This term is also used when the index sets are mathematical spaces other than the real line,<ref name="GikhmanSkorokhod1969page1"/><ref name="GusakKukush2010page1">{{harvtxt|Gusak|Kukush|Kulik|Mishura|2010}}, p. 1</ref> while the terms ''stochastic process'' and ''random process'' are usually used when the index set is interpreted as time,<ref name="GikhmanSkorokhod1969page1"/><ref name="GusakKukush2010page1"/><ref name="Bass2011page1">{{cite book|author=Richard F. Bass|title=Stochastic Processes|url=https://books.google.com/books?id=Ll0T7PIkcKMC|year=2011|publisher=Cambridge University Press|isbn=978-1-139-50147-7|page=1}}</ref> 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]].<ref name="GikhmanSkorokhod1969page1"/><ref name="Lamperti1977page1"/><ref name="AdlerTaylor2009page7"/>

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>.

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>.

在概率论中起着核心作用的'''<font color="#ff8000"> 维纳过程Wiener process</font>'''，通常被认为是最重要的和研究过的随机过程，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<math>n</math>维的'''<font color="#ff8000"> 欧氏空间Euclidean space</font>'''。如果增量的平均值为零，那么由此产生的维纳Wiener或布朗Brownian运动过程称为具有零漂移。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<math> \mu</math>，即一个实数，那么得到的随机过程就具有<math> \mu</math>漂移。

在概率论中起着核心作用的'''<font color="#ff8000"> 维纳过程</font>'''，通常被认为是最重要的和研究过的随机过程，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<math>n</math>维的'''<font color="#ff8000"> 欧氏空间Euclidean space</font>'''。如果增量的平均值为零，那么由此产生的维纳Wiener或布朗Brownian运动过程称为具有零漂移。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<math> \mu</math>，即一个实数，那么得到的随机过程就具有<math> \mu</math>漂移。

===Notation符号===

===Notation符号===

A stochastic process can be denoted, among other ways, by <math>\{X(t)\}_{t\in T} </math>,<ref name="Brémaud2014page120"/> <math>\{X_t\}_{t\in T} </math>,<ref name="Asmussen2003page408"/> <math>\{X_t\}</math><ref name="Lamperti1977page3">,{{cite book|author=John Lamperti|title=Stochastic processes: a survey of the mathematical theory|url=https://books.google.com/books?id=Pd4cvgAACAAJ|year=1977|publisher=Springer-Verlag|isbn=978-3-540-90275-1|page=3}}</ref> <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 notation#Function notation|abuse of function notation]].<ref name="Klebaner2005page55">{{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=55}}</ref> 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.<ref name="Lamperti1977page3"/> If the index set is <math>T=[0,\infty)</math>, then one can write, for example, <math>(X_t , t \geq 0)</math> to denote the stochastic process.<ref name="ChaumontYor2012"/>

A stochastic process can be denoted, among other ways, by <math>\{X(t)\}_{t\in T} </math>,<ref name="Brémaud2014page120"/> <math>\{X_t\}_{t\in T} </math>,<ref name="Asmussen2003page408"/> <math>\{X_t\}</math><ref name="Lamperti1977page3">,{{cite book|author=John Lamperti|title=Stochastic processes: a survey of the mathematical theory|url=https://books.google.com/books?id=Pd4cvgAACAAJ|year=1977|publisher=Springer-Verlag|isbn=978-3-540-90275-1|page=3}}</ref> <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 notation#Function notation|abuse of function notation]].<ref name="Klebaner2005page55">{{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=55}}</ref> 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.<ref name="Lamperti1977page3"/> If the index set is <math>T=[0,\infty)</math>, then one can write, for example, <math>(X_t , t \geq 0)</math> to denote the stochastic process.<ref name="ChaumontYor2012"/>

随机过程可以用<math>\{X（t）\{t}</math>，<ref name=“Brémaud2014page120”/><math>\{X\{t\}在t}</math>中，<ref name=“Asmussen2003page408”/><math>\{X\}</math><ref name=“Lamperti1977page3”>，{引用书籍{作者=John Lamperti | title=随机过程：数学理论综述=https://books.google.com/books？id=pd4cvgaacaj | year=1977 | publisher=Springer Verlag | isbn=978-3-540-90275-1 | page=3}</ref><math>\{X（t）\}</math>或简单地称为<math>X</math>或<math>X（t）</math>，尽管<math>X（t）</math>被视为[[符号滥用#函数表示法|函数表示法滥用]]。<ref name=“Klebaner2005page55”>{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=55}</ref>例如，<math>X（t）</math>或<math>X_t</math>引用具有索引<math>t</math>的随机变量，而不是整个随机过程。<ref name=“Lamperti1977page3”/>如果索引集是<math>t=[0，\infty）</math>，然后，我们可以写，例如，<math>（X\u t，t\geq 0）</math>来表示随机过程。<ref name=“ChaumontYor2012”/>

随机过程可以用<math>\{X（t）\{t}</math>，<math>\{X（t）\}</math><ref name="Brémaud2014page120"/> <math>\{X_t\}_{t\in T} </math>,<ref name="Asmussen2003page408"/> <math>\{X_t\}</math><ref name="Lamperti1977page3">,{{cite book|author=John Lamperti|title=Stochastic processes: a survey of the mathematical theory|url=https://books.google.com/books?id=Pd4cvgAACAAJ|year=1977|publisher=Springer-Verlag|isbn=978-3-540-90275-1|page=3}}</ref>或简单地称为<math>X</math>或<math>X（t）</math>，尽管<math>X（t）</math>被视为[[符号滥用#函数表示法|函数表示法滥用]]。<ref name="Klebaner2005page55">{{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=55}}</ref> 例如， <math>X(t)</math> 或 <math>X_t</math>引用具有索引<math>t</math>的随机变量，而不是整个随机过程。<ref name="Lamperti1977page3"/>如果索引集是<math>T=[0,\infty)</math>，然后，我们可以写，例如，<math>(X_t , t \geq 0)</math>来表示随机过程。<ref name=“ChaumontYor2012”/>

Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered as a continuous version of the simple random walk. The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled, which is the subject of Donsker's theorem or invariance principle, also known as the functional central limit theorem.

Almost surely, a sample path of a Wiener process is continuous everywhere but nowhere differentiable. It can be considered as a continuous version of the simple random walk. The process arises as the mathematical limit of other stochastic processes such as certain random walks rescaled, which is the subject of Donsker's theorem or invariance principle, also known as the functional central limit theorem.

几乎可以肯定，'''<font color="#ff8000"> 维纳过程Wiener process</font>'''的样本路径在任何地方都是连续的，但是没有可微的地方。它可以看作是简单随机游走的连续形式。这个过程作为其他随机过程的数学极限出现，例如某些随机游动的重新标度，这是 Donsker 定理或不变性原理的主题，也被称为函数中心极限定理。

几乎可以肯定，'''<font color="#ff8000"> 维纳过程Wiener process</font>'''的样本路径在任何地方都是连续的，但是没有可微的地方。它可以看作是简单随机游走的连续形式。这个过程作为其他随机过程的数学极限出现，例如某些随机游走的重新标度，这是 Donsker 定理或不变性原理的主题，也被称为<font color="#ff8000"> 函数中心极限定理</font>。

==Examples示例==

==Examples示例==