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===索引集 Index set===

===索引集 Index set===

The set <math>T</math> is called the '''index set'''<ref name="Parzen1999"/><ref name="Florescu2014page294"/> or '''parameter set'''<ref name="Lamperti1977page1"/><ref name="Skorokhod2005page93">{{cite book|author=Valeriy Skorokhod|title=Basic Principles and Applications of Probability Theory|url=https://books.google.com/books?id=dQkYMjRK3fYC|year=2005|publisher=Springer Science & Business Media|isbn=978-3-540-26312-8|pages=93, 94}}</ref> of the stochastic process. Often this set is some subset of the [[real line]], such as the [[natural numbers]] or an interval, giving the set <math>T</math> the interpretation of time.<ref name="doob1953stochasticP46to47"/> In addition to these sets, the index set <math>T</math> can be other linearly ordered sets or more general mathematical sets,<ref name="doob1953stochasticP46to47"/><ref name="Billingsley2008page482">{{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=482}}</ref> such as the Cartesian plane <math>R^2</math> or <math>n</math>-dimensional Euclidean space, where an element <math>t\in T</math> can represent a point in space.<ref name="KarlinTaylor2012page27">{{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=27}}</ref><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=25}}</ref> But in general more results and theorems are possible for stochastic processes when the index set is ordered.<ref name="Skorokhod2005page104">{{cite book|author=Valeriy Skorokhod|title=Basic Principles and Applications of Probability Theory|url=https://books.google.com/books?id=dQkYMjRK3fYC|year=2005|publisher=Springer Science & Business Media|isbn=978-3-540-26312-8|page=104}}</ref>

集合<math>T</math>称为“索引集”<ref name=“Parzen1999”/><ref name=“Florescu2014page294”/>或“参数集”<ref name="Lamperti1977page1"/><ref name="Skorokhod2005page93">{{cite book|author=Valeriy Skorokhod|title=Basic Principles and Applications of Probability Theory|url=https://books.google.com/books?id=dQkYMjRK3fYC|year=2005|publisher=Springer Science & Business Media|isbn=978-3-540-26312-8|pages=93, 94}}</ref>。通常，这个集合是实线的一个子集，例如自然数或一个区间，使集合<math>T</math>能够解释时间。<ref name=“doob1953stochasticP46to47”/>除了这些集合，索引集<math>T</math>可以是其他线性有序集或更一般的数学集，<ref name="doob1953stochasticP46to47"/><ref name="Billingsley2008page482">{{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=482}}</ref>例如笛卡尔平面<math>R^2</math>或<math>n</math>维欧几里得空间，其中t中的元素可以表示空间中的一个点。<ref name="KarlinTaylor2012page27">{{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=27}}</ref><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=25}}</ref>但一般情况下，当索引集有序时，随机过程可以得到更多的结果和定理。<ref name="Skorokhod2005page104">{{cite book|author=Valeriy Skorokhod|title=Basic Principles and Applications of Probability Theory|url=https://books.google.com/books?id=dQkYMjRK3fYC|year=2005|publisher=Springer Science & Business Media|isbn=978-3-540-26312-8|page=104}}</ref>

 

集合<math>T</math>称为“索引集”<ref name=“Parzen1999”/><ref name=“Florescu2014page294”/>或“‘参数集’”<ref name=“Lamperti1977page1”/><ref name=“Skorokhod2005page93”>{cite book | author=Valeriy skorokord | title=概率论的基本原理和应用=https://books.google.com/books？随机过程的id=dQkYMjRK3fYC | year=2005 | publisher=Springer Science&Business Media | isbn=978-3-540-26312-8 | pages=93，94}}</ref>。通常，这个集合是[[实线]]的一个子集，例如[[自然数]]或一个区间，使集合<math>T</math>能够解释时间。<ref name=“doob1953stochasticP46to47”/>除了这些集合，索引集<math>T</math>可以是其他线性有序集或更一般的数学集，<ref name=“doob1953stochasticP46to47”/><ref name=“Billingsley2008page482”>{cite book | author=Patrick Billingsley | title=Probability and Measure |网址=https://books.google.com/books？id=qyxqoxyeic | year=2008 | publisher=Wiley India Pvt.Limited | isbn=978-81-265-1771-8 | page=482}}</ref>例如笛卡尔平面<math>R^2</math>或<math>n</math>维欧几里得空间，其中t中的元素可以表示空间中的一个点=https://books.google.com/books？id=dSDxjX9nmmMC | year=2012 | publisher=academical Press | isbn=978-0-08-057041-9 | page=27}</ref>{cite book | author1=Donald L.Snyder | author2=Michael I.Miller | title=时空中的随机点过程| url=https://books.google.com/books？id=c_3UBwAAQBAJ | year=2012 | publisher=Springer Science&Business Media | isbn=978-1-4612-3166-0 | page=25}</ref>但一般情况下，当索引集有序时，随机过程可以得到更多的结果和定理。<ref name=“skorokod2005page104”>{cite book | author=Valeriy skorokorokod | title=概率的基本原理和应用理论|网址=https://books.google.com/books？id=dQkYMjRK3fYC |年=2005 | publisher=Springer Science&Business Media | isbn=978-3-540-26312-8 | page=104}</ref>

===状态空间 State space ===

===状态空间 State space ===