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

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>'''的例子被称为简单随机游走，这是一个以整数为状态空间的离散时间随机过程，它基于一个伯努利过程，其中每个 Bernoulli 变量要么取值为正，要么取值为负。换句话说，简单随机游动发生在整数上，它的值随概率的增加而增加1，或随概率的减少而减少1，所以这种'''<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>'''。

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>