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随机过程 (查看源代码)

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Martingales have many applications in statistics, but it has been remarked that its use and application are not as widespread as it could be in the field of statistics, particularly statistical inference.<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> They have found applications in areas in probability theory such as queueing theory and Palm calculus<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> and other fields such as economics<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"/>

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'''鞅Martingale'''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，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微积分与随机循环~~=https://books.google.com/books？id=dh3pcaaqbaj&pg=PR2 | year=2013 | publisher=~~Springer Science&Business Media~~ | isbn=978-3-662-11657-9}</ref>~~和其他领域，如经济学~~<ref name=“HallHeyde2014pageX”{cite book | author1=P.Hall | author2=C.C.Heyde | title=鞅极限理论及其应用| url=https://books.google.com/books？id=gqriBQAAQBAJ&pg=PR10 | ~~year~~=2014 | publisher=Elsevier Science | isbn=978-1-4832-6322-9 | page=x}</ref>~~和财务。~~<ref name=“Musielarukowski2006”/>

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'''鞅Martingale'''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，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=排队论的元素：Palm鞅演算和随机递归| url=https://books.google.com/books？id=dh3pcaaqbaj&pg=PR2 | year=2013 | publisher=Springer科学与商业媒体| 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=鞅极限理论及其应用| url=https://books.google.com/books？id=gqriBQAAQBAJ&pg=PR10 |年份=2014 | publisher=Elsevier Science | isbn=978-1-4832-6322-9 | page=x}</ref>和金融。<ref name=“Musielarukowski2006”/>

===Lévy process莱维过程===

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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

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~~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

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''' 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

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继卡达诺之后，[[Jakob Bernoulli]]{{efn |也被称为杰姆斯或杰姆斯伯努利Jacques Bernoulli<ref name=“Hald2005page221”>{cite book | author=Anders Hald | title=1750年前概率统计及其应用的历史=https://books.google.com/books？id=pOQy6-qnVx8C |年份=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”>{引用书| author=L.E.Maistrov | title=概率论：历史素描|网址=https://books.google.com/books？id=2ZbiBQAAQBAJ&pg=PR9 |年份=2014 | publisher=Elsevier Science | isbn=978-1-4832-1863-2 | page=56}</ref><ref name=“Tabak2014page37”>{cite book | author=John Tabak | title=概率与统计学：不确定性科学|网址=https://books.google.com/books？id=h3WVqBPHboAC |年=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”>{引用期刊| last1=Chung | first1=Kai Lai | title=Probability and Doob | journal=The American Mathematic Monthly | volume=105 | isson=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=概率统计史研究XLVI。概率度量：从Lebesgue到Kolmogorov | journal=Biometrika | volume=87 | Isse=1 | year=2000 | pages=145-156 | issn=0006-3444 | doi=10.1093/biomet/87.1.145}</ref>，大多数数学界人士都注意到，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”>{{引用期刊| last1=Benzi | first1=Margherita | last2=Benzi | first2=Michele | last3=Seneta | first3=Eugene | title=Francesco Paolo Cantelli。b、 1875年12月20日d.1966年7月21日|期刊=国际统计评论|卷=75 |问题=2 |年=2007 |页=128 | issn=0306-7734 | doi=10.1111/j.1751-5823.2007.00009.x}}</ref>}}直到20世纪，概率论才被认为是数学的一部分。<ref name=“Chun1998年”/><ref name=“BenziBenzi2007年”/><ref name=“DoobB1996”>{{cite journal | last1=Doob | first1=Joseph L.；title=数学概率中严谨性的发展（1900-1950年）| journal=美国数学月刊|卷=103 |问题=7 |页=586–595 |年=1996年| issn=0002-9890 | doi=10.2307/2974673 | jstor=2974673 }}</ref><ref name=“Cramer1976”>{cite journal | last1=Cramer | first1=Harald | title=半个世纪与概率论：一些个人回忆| journal=The annalls of Probability | volume=4 | issn=4 | year=1976 | pages=509-546 | issn=0091-1798 | doi=10.1214/aop/117696025 | doi access=free}</ref>

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===Statistical mechanics统计力学===

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