561
个编辑
更改
→Stochastic process随机过程
<center><math>
<center><math>
<center><math>
<center><math>
</math></center>
</math></center>
</math></center>
</math></center>
where <math>n\geq 1</math> is a counting number and each set <math>t_i</math> is a non-empty finite subset of the index set <math>T</math>, so each <math>t_i\subset T</math>, which means that <math>t_1,\dots,t_n</math> is any finite collection of subsets of the index set <math>T</math>.
where <math>n\geq 1</math> is a counting number and each set <math>t_i</math> is a non-empty finite subset of the index set <math>T</math>, so each <math>t_i\subset T</math>, which means that <math>t_1,\dots,t_n</math> is any finite collection of subsets of the index set <math>T</math>.
其中 < math > n geq 1 </math > 是一个计数数字，每个集 < math > t i </math > 是指数集 < math > t </math > 的非空有限子集，因此每个 < math > t i 子集 t </math > ，这意味着 < math > t _ 1，点，t _ n </math > 是指数集 < math > t </math > 的任何有限子集。
其中 <math>n\geq 1</math> 是一个计数数字，每个集 <math>t_i</math> 是指数集 <math>T</math> 的非空有限子集，因此每个 <math>t_i\subset T</math> ，这意味着 <math>t_1,\dots,t_n</math>是指数集 <math>T</math> 的任何有限子集。
Historically, in many problems from the natural sciences a point <math>t\in T</math> had the meaning of time, so <math>X(t)</math> is a random variable representing a value observed at time <math>t</math>.<ref name="Borovkov2013page528">{{cite book|author=Alexander A. Borovkov|authorlink=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=528}}</ref> A stochastic process can also be written as <math> \{X(t,\omega):t\in T \}</math> to reflect that it is actually a function of two variables, <math>t\in T</math> and <math>\omega\in \Omega</math>.<ref name="Lamperti1977page1"/><ref name="LindgrenRootzen2013page11">{{cite book|author1=Georg Lindgren|author2=Holger Rootzen|author3=Maria Sandsten|title=Stationary Stochastic Processes for Scientists and Engineers|url=https://books.google.com/books?id=FYJFAQAAQBAJ&pg=PR1|year=2013|publisher=CRC Press|isbn=978-1-4665-8618-5|pages=11}}</ref>
Historically, in many problems from the natural sciences a point <math>t\in T</math> had the meaning of time, so <math>X(t)</math> is a random variable representing a value observed at time <math>t</math>.<ref name="Borovkov2013page528">{{cite book|author=Alexander A. Borovkov|authorlink=Alexander A. Borovkov|title=Probability Theory|url=https://books.google.com/books?id=hRk_AAAAQBAJ&pg|year=2013|publisher=Springer Science & Business Media|isbn=978-1-4471-5201-9|page=528}}</ref> A stochastic process can also be written as <math> \{X(t,\omega):t\in T \}</math> to reflect that it is actually a function of two variables, <math>t\in T</math> and <math>\omega\in \Omega</math>.<ref name="Lamperti1977page1"/><ref name="LindgrenRootzen2013page11">{{cite book|author1=Georg Lindgren|author2=Holger Rootzen|author3=Maria Sandsten|title=Stationary Stochastic Processes for Scientists and Engineers|url=https://books.google.com/books?id=FYJFAQAAQBAJ&pg=PR1|year=2013|publisher=CRC Press|isbn=978-1-4665-8618-5|pages=11}}</ref>
历史上，在许多自然科学问题中，一个点<math>t\in T</math> 具有时间的意义，因此，<math>X(t)</math>表示是一个在时间<math>t</math>的随机变量。<ref name=“Borovkov2013page528”>{cite book | authorlink=Alexander a.Borovkov | title=Probability Theory | url=https://books.google.com/books？id=hRk_AAAAQBAJ&pg | year=2013 | publisher=Springer Science&Business Media | isbn=978-1-4471-5201-9 | page=528}</ref>随机过程也可以写成<math>\{X（t，omega）：t\ in t\}</math>来反映它实际上是两个变量的函数，<math>t\in t</math>和<math>\omega\in\omega</math><ref name=“Lamperti1977page1”/><ref name=“LindgrenRootzen2013page11”>{cite book | author1=Georg Lindgren | author2=Holger Rootzen | author3=Maria Sandsten | title=科学家和工程师的平稳随机过程| url=https://books.google.com/books？id=fyjfaqbaj&pg=PR1 | year=2013 | publisher=CRC出版社| isbn=978-1-4665-8618-5 | pages=11}</ref>
For any measurable subset <math>C</math> of the <math>n</math>-fold Cartesian power <math>S^n=S\times\dots \times S</math>, the finite-dimensional distributions of a stochastic process <math>X</math> can be written as: But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.
For any measurable subset <math>C</math> of the <math>n</math>-fold Cartesian power <math>S^n=S\times\dots \times S</math>, the finite-dimensional distributions of a stochastic process <math>X</math> can be written as: But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.