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随机过程 (查看源代码)

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→‎样本函数 Sample function

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===样本函数 Sample function===

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'''样本函数'''是随机过程的单个结果，因此，它是由随机过程中每个随机变量的一个可能值构成的。<ref name="Lamperti1977page1"/><ref name="Florescu2014page296">{{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=296}}</ref>更准确地说，如果<math>\{X（t，omega）：t\in t\}</math>是一个随机过程，那么对于任何点<math>\omega\in\omega</math>，映射

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A '''~~sample function~~''' is a single [[Outcome (probability)|outcome]] of a stochastic process, so it is formed by taking a single possible value of each random variable of the stochastic process. More precisely, if <math>\{X(t,\omega):t\in T \}</math> is a stochastic process, then for any point <math>\omega\in\Omega</math>, the [[Map (mathematics)|mapping]]

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~~“样本函数”是随机过程的单个结果，因此，它是由随机过程中每个随机变量的一个可能值构成的。~~<ref name="Lamperti1977page1"/><ref name="Florescu2014page296">{{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=296}}</ref>更准确地说，如果<math>\{X（t，omega）：t\in t\}</math>是一个随机过程，那么对于任何点<math>\omega\in\omega</math>，映射

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</math></center>

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is called a sample function, a '''realization''', or, particularly when <math>T</math> is interpreted as time, a '''sample path''' of the stochastic process <math>\{X(t,\omega):t\in T \}</math>. This means that for a fixed <math>\omega\in\Omega</math>, there exists a sample function that maps the index set <math>T</math> to the state space <math>S</math>.<ref name="Lamperti1977page1"/> Other names for a sample function of a stochastic process include '''trajectory''', '''path function'''<ref name="Billingsley2008page493">{{cite book|author=Patrick Billingsley|title=Probability and Measure|url=https://books.google.com/books?id=QyXqOXyxEeIC|year=2008|publisher=Wiley India Pvt. Limited|isbn=978-81-265-1771-8|page=493}}</ref> or '''path'''.<ref name="Øksendal2003page10">{{cite book|author=Bernt Øksendal|title=Stochastic Differential Equations: An Introduction with Applications|url=https://books.google.com/books?id=VgQDWyihxKYC|year=2003|publisher=Springer Science & Business Media|isbn=978-3-540-04758-2|page=10}}</ref>

称为样本函数，称为“实现”，或者，特别是当<math>T</math>被解释为时间时，随机过程的“样本路径”<math>\{X（T，omega）：T\in T\}</math>。<ref name="RogersWilliams2000page121b">{{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|pages=121–124}}</ref>这意味着对于一个固定的<math>\omega\in\omega</math>，存在一个将索引集<math>T</math>映射到状态空间<math>S</math><ref name="Lamperti1977page1"/> 的示例函数的其他名称随机过程包括“轨迹”、“路径函数”<ref name="Billingsley2008page493">{{cite book|author=Patrick Billingsley|title=Probability and Measure|url=https://books.google.com/books?id=QyXqOXyxEeIC|year=2008|publisher=Wiley India Pvt. Limited|isbn=978-81-265-1771-8|page=493}}</ref>或“路径”.<ref name="Øksendal2003page10">{{cite book|author=Bernt Øksendal|title=Stochastic Differential Equations: An Introduction with Applications|url=https://books.google.com/books?id=VgQDWyihxKYC|year=2003|publisher=Springer Science & Business Media|isbn=978-3-540-04758-2|page=10}}</ref>

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