随机过程 - 版本历史

2022年3月12日 (六) 00:41 Fernando

2022-03-12T00:41:54Z

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Fernando：almost done

2022-03-11T22:40:53Z

almost done

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Fernando：编辑到维纳过程

2022-01-14T13:02:13Z

编辑到维纳过程

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Fernando：到Introduction的分类

2021-12-19T10:53:05Z

到Introduction的分类

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薄荷：/* 编辑推荐 */

2021-10-29T14:43:42Z

编辑推荐

薄荷：/* 编辑推荐 */

2021-10-29T14:43:05Z

编辑推荐

薄荷：/* 早期概率论 Early probability theory */

2021-10-29T14:39:30Z

早期概率论 Early probability theory

薄荷：/* 点过程 Point process */

2021-10-29T14:38:38Z

点过程 Point process

薄荷：/* 维纳过程 Wiener process */

2021-10-29T14:37:52Z

维纳过程 Wiener process

2021年10月29日 (五) 14:35 薄荷

2021-10-29T14:35:49Z

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←上一版本		2021年10月29日 (五) 14:43的版本
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−	~~本中文词条由tueryeye翻译和审校，~~[[用户:薄荷\|薄荷]]编辑，如有问题，欢迎在讨论页面留言。	+	本中文词条水流心不竞翻译，[[用户:薄荷\|薄荷]]编辑，如有问题，欢迎在讨论页面留言。

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 ==编辑推荐==
 ===集智课程===
-====[]====
+====[https://campus.swarma.org/course/3157 熵率（Entropy-Rate）]====

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 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>
-但尽管一些著名的数学家对概率论做出了贡献，比如[[皮埃尔-西蒙-拉普拉斯]、[[亚伯拉罕-德-莫伊夫]]、[[卡尔-高斯]]、[[西蒙-泊阿松Siméon Poisson]]和[[帕夫努蒂·切比雪夫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>指导20世纪，大多数数学界人士（一个显著的例外是俄罗斯的圣彼得堡学派，在那里，以切比雪夫为首的数学家研究概率论<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>）才认为概率论是数学的一部分。<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>
+但尽管一些著名的数学家对概率论做出了贡献，比如[[皮埃尔-西蒙-拉普拉斯]、[[亚伯拉罕-德-莫伊夫]]、[[卡尔-高斯]]、[[西蒙-泊阿松]] Siméon Poisson和[[帕夫努蒂·切比雪夫]] 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>指导20世纪，大多数数学界人士（一个显著的例外是俄罗斯的圣彼得堡学派，在那里，以切比雪夫为首的数学家研究概率论<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>）才认为概率论是数学的一部分。<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>
 ===统计力学 Statistical mechanics===

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 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"/>
 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>
 ==历史 History==

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 维纳过程是一个随机过程，具有平稳的独立的增量并且基于增量的大小是正态分布的.<ref name="RogersWilliams2000page1">{{cite book|author1=L. C. G. Rogers|author2=David Williams|title=Diffusions, Markov Processes, and Martingales: Volume 1, Foundations|url=https://books.google.com/books?id=W0ydAgAAQBAJ&pg=PA1|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|page=1}}</ref><ref name="Klebaner2005page56">{{cite book|author=Fima C. Klebaner|title=Introduction to Stochastic Calculus with Applications|url=https://books.google.com/books?id=JYzW0uqQxB0C|year=2005|publisher=Imperial College Press|isbn=978-1-86094-555-7|page=56}}</ref>维纳过程是以Norbert Wiener命名的，他证明了它的数学存在性，但是这个过程也被称为布朗运动过程或仅仅是布朗运动，因为它是液体中[[布朗运动]]的模型。<ref name="Brush1968page1">{{cite journal|last1=Brush|first1=Stephen G.|title=A history of random processes|journal=Archive for History of Exact Sciences|volume=5|issue=1|year=1968|pages=1–2|issn=0003-9519|doi=10.1007/BF00328110}}</ref><ref name="Applebaum2004page1338">{{cite journal|last1=Applebaum|first1=David|title=Lévy processes: From probability to finance and quantum groups|journal=Notices of the AMS|volume=51|issue=11|year=2004|pages=1338}}</ref><ref name="Applebaum2004page1338"/><ref name="GikhmanSkorokhod1969page21">{{cite book|author1=Iosif Ilyich Gikhman|author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=yJyLzG7N7r8C&pg=PR2|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=21}}</ref>
-[[File:DriftedWienerProcess1D.svg|thumb|left|实现维纳Wiener过程（或布朗运动过程），具有漂移（<font color=blue>蓝色</font>）且不漂移（<font color=red>红色</font>）。]]
+[[File:维纳过程.png|thumb|left|实现维纳Wiener过程（或布朗运动过程），具有漂移（<font color=blue>蓝色</font>）且不漂移（<font color=red>红色</font>）。]]
 ''' Wiener process维纳过程'''在概率论中起着中心作用，通常被认为是最重要和研究的随机过程，并与其他随机过程联系在一起<ref name="doob1953stochasticP46to47"/><ref name="RogersWilliams2000page1"/><ref name="Steele2012page29">{{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=29}}</ref><ref name="Florescu2014page471">{{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=471}}</ref><ref name="KarlinTaylor2012page21">{{cite book|author1=Samuel Karlin|author2=Howard E. Taylor|title=A First Course in Stochastic Processes|url=https://books.google.com/books?id=dSDxjX9nmmMC|year=2012|publisher=Academic Press|isbn=978-0-08-057041-9|pages=21, 22}}</ref><ref name="KaratzasShreve2014pageVIII">{{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=VIII}}</ref><ref name="RevuzYor2013pageIX">{{cite book|author1=Daniel Revuz|author2=Marc Yor|title=Continuous Martingales and Brownian Motion|url=https://books.google.com/books?id=OYbnCAAAQBAJ|year=2013|publisher=Springer Science & Business Media|isbn=978-3-662-06400-9|page=IX}}</ref>其索引集和状态空间分别是非负数和实数，因此它既有连续索引集又有状态空间<ref name="Rosenthal2006page186">{{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|page=186}}</ref>但是过程可以定义得更广泛，这样它的状态空间可以是维欧几里德空间。<ref name="Klebaner2005page81"/><ref name="KarlinTaylor2012page21"/><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=33}}</ref>如果任何增量的[[平均值]]为零，则所得到的维纳或布朗运动过程称为零漂移。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>，即实数，由此产生的随机过程被称为'''漂移'''。<ref name="Steele2012page118">{{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=118}}</ref><ref name="MörtersPeres2010page1"/><ref name="KaratzasShreve2014page78">{{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=78}}</ref>