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→随机游走 Random walk
随机游走是随机过程，通常定义为欧几里德空间中独立同分布的随机变量随机变量或随机向量的和，因此它们是离散时间变化的过程。<ref name="Klenke2013page347">{{cite book|author=Achim Klenke|title=Probability Theory: A Comprehensive Course|url=https://books.google.com/books?id=aqURswEACAAJ|year=2013|publisher=Springer|isbn=978-1-4471-5362-7|pages=347}}</ref><ref name="LawlerLimic2010page1">{{cite book|author1=Gregory F. Lawler|author2=Vlada Limic|title=Random Walk: A Modern Introduction|url=https://books.google.com/books?id=UBQdwAZDeOEC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48876-1|page=1}}</ref><ref name="Kallenberg2002page136">{{cite book|author=Olav Kallenberg|title=Foundations of Modern Probability|url=https://books.google.com/books?id=L6fhXh13OyMC|date= 2002|publisher=Springer Science & Business Media|isbn=978-0-387-95313-7|page=136}}</ref><ref name="Florescu2014page383">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|page=383}}</ref><ref name="Durrett2010page277">{{cite book|author=Rick Durrett|title=Probability: Theory and Examples|url=https://books.google.com/books?id=evbGTPhuvSoC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-49113-6|page=277}}</ref>但是有些人也使用这个术语来指代连续时间变化的过程，<ref name=“Weiss2006page1”>{cite book | last1=Weiss | first1=George H.| title=Statistical Sciences | chapter=Random Walks | year=2006 | doi=10.1002/0471667196.ess2180.pub2 | page=1 | isbn=978-0471667193}}</ref>尤其是金融中使用的维纳过程，这导致了一些混乱，导致其受到批评。<ref name="Spanos1999page454">{{cite book|author=Aris Spanos|title=Probability Theory and Statistical Inference: Econometric Modeling with Observational Data|url=https://books.google.com/books?id=G0_HxBubGAwC|year=1999|publisher=Cambridge University Press|isbn=978-0-521-42408-0|page=454}}</ref>还有其他各种类型的随机游动，它们的状态空间可以是其他数学对象，例如格和群，一般来说，它们都是高度研究的，在不同的学科中有许多应用。<ref name="Weiss2006page1"/><ref name="Klebaner2005page81">{{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=81}}</ref>
随机游走是随机过程，通常定义为欧几里德空间中独立同分布的随机变量随机变量或随机向量的和，因此它们是离散时间变化的过程。<ref name="Klenke2013page347">{{cite book|author=Achim Klenke|title=Probability Theory: A Comprehensive Course|url=https://books.google.com/books?id=aqURswEACAAJ|year=2013|publisher=Springer|isbn=978-1-4471-5362-7|pages=347}}</ref><ref name="LawlerLimic2010page1">{{cite book|author1=Gregory F. Lawler|author2=Vlada Limic|title=Random Walk: A Modern Introduction|url=https://books.google.com/books?id=UBQdwAZDeOEC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-48876-1|page=1}}</ref><ref name="Kallenberg2002page136">{{cite book|author=Olav Kallenberg|title=Foundations of Modern Probability|url=https://books.google.com/books?id=L6fhXh13OyMC|date= 2002|publisher=Springer Science & Business Media|isbn=978-0-387-95313-7|page=136}}</ref><ref name="Florescu2014page383">{{cite book|author=Ionut Florescu|title=Probability and Stochastic Processes|url=https://books.google.com/books?id=Z5xEBQAAQBAJ&pg=PR22|year=2014|publisher=John Wiley & Sons|isbn=978-1-118-59320-2|page=383}}</ref><ref name="Durrett2010page277">{{cite book|author=Rick Durrett|title=Probability: Theory and Examples|url=https://books.google.com/books?id=evbGTPhuvSoC|year=2010|publisher=Cambridge University Press|isbn=978-1-139-49113-6|page=277}}</ref>但是有些人也使用这个术语来指代连续时间变化的过程，<ref name=“Weiss2006page1”>{cite book | last1=Weiss | first1=George H.| title=Statistical Sciences | chapter=Random Walks | year=2006 | doi=10.1002/0471667196.ess2180.pub2 | page=1 | isbn=978-0471667193}}</ref>尤其是金融中使用的维纳过程，这导致了一些混乱，导致其受到批评。<ref name="Spanos1999page454">{{cite book|author=Aris Spanos|title=Probability Theory and Statistical Inference: Econometric Modeling with Observational Data|url=https://books.google.com/books?id=G0_HxBubGAwC|year=1999|publisher=Cambridge University Press|isbn=978-0-521-42408-0|page=454}}</ref>还有其他各种类型的随机游动，它们的状态空间可以是其他数学对象，例如格和群，一般来说，它们都是高度研究的，在不同的学科中有许多应用。<ref name="Weiss2006page1">Weiss, George H. (2006). "Random Walks". Encyclopedia of Statistical Sciences. p. 1. doi:10.1002/0471667196.ess2180.pub2. ISBN 978-0471667193.</ref><ref name="Klebaner2005page81">{{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=81}}</ref>
随机游走的一个经典例子被称为“简单随机游动”，它是一个离散时间的随机过程，以整数为状态空间，它基于伯努利过程，其中每个贝努利变量取正值或负值。换言之，简单随机游走发生在整数上，例如其值随概率<math>p</math>增加1，，或随着概率<math>1-p</math>而减小1，因此这种随机游动的指标集是自然数，而其状态空间是整数。如果<math>p=0.5</math>，这种随机游动称为对称随机游动。<ref name="Gut2012page88">{{cite book|author=Allan Gut|title=Probability: A Graduate Course|url=https://books.google.com/books?id=XDFA-n_M5hMC|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4614-4708-5|page=88}}</ref><ref name="GrimmettStirzaker2001page71">{{cite book|author1=Geoffrey Grimmett|author2=David Stirzaker|title=Probability and Random Processes|url=https://books.google.com/books?id=G3ig-0M4wSIC|year=2001|publisher=OUP Oxford|isbn=978-0-19-857222-0|page=71}}</ref>
随机游走的一个经典例子被称为“简单随机游动”，它是一个离散时间的随机过程，以整数为状态空间，它基于伯努利过程，其中每个贝努利变量取正值或负值。换言之，简单随机游走发生在整数上，例如其值随概率<math>p</math>增加1，，或随着概率<math>1-p</math>而减小1，因此这种随机游动的指标集是自然数，而其状态空间是整数。如果<math>p=0.5</math>，这种随机游动称为对称随机游动。<ref name="Gut2012page88">{{cite book|author=Allan Gut|title=Probability: A Graduate Course|url=https://books.google.com/books?id=XDFA-n_M5hMC|year=2012|publisher=Springer Science & Business Media|isbn=978-1-4614-4708-5|page=88}}</ref><ref name="GrimmettStirzaker2001page71">{{cite book|author1=Geoffrey Grimmett|author2=David Stirzaker|title=Probability and Random Processes|url=https://books.google.com/books?id=G3ig-0M4wSIC|year=2001|publisher=OUP Oxford|isbn=978-0-19-857222-0|page=71}}</ref>
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===维纳过程 Wiener process===
===维纳过程 Wiener process===