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'distributions of a stochastic process'
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| Title of object: |
distributions of a stochastic process |
| Canonical Name: |
DistributionsOfAStochasticProcess |
| Type: |
Definition |
| Created on: |
2005-06-22 14:46:06 |
| Modified on: |
2005-06-22 22:54:48 |
| Classification: |
msc:60G07 |
| Defines: |
finite dimensional distribution, f.f.d., identically distributed stochastic process, version of a stochastic process |
| Synonyms: |
distributions of a stochastic process=finite dimensional probability distribution
distributions of a stochastic process=ffd |
Preamble:
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Content:
Just as one can associate a random variable $X$ with its distribution $F_X$, one can associate a stochastic process $\lbrace X(t) \mid t\in T \rbrace$ with some distributions, such that the distributions will more or less describe the process. While the set of distributions $\lbrace F_{X(t)} \mid t\in T \rbrace$ can describe the random variables $X(t)$ individually, it says nothing about the
relationships between any pair, or more generally, any finite set of random variables $X(t)$'s at different $t$'s.
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To resolve this, we enlarge the above set to include all joint probability distributions of finitely many $X(t)$'s, called \emph{the family of finite dimensional probability distributions}.Specifically, let $n<\infty$ be any positive integer, an \emph{$n$-dimensional probability distribution} of the stochastic process $\lbrace X(t) \mid t\in T \rbrace$ is a joint probability
distribution of $X(t_1),\ldots,X(t_n)$, where $t_i\in T$:
$$F_{t_1,\ldots,t_n}(x_1,\ldots,x_n):=F_{X(t_1),\ldots,X(t_n)}(x_1,\ldots,x_n)
=P(\lbrace X(t_1)\leq x_1 \rbrace \cap \cdots \cap \lbrace
X(t_n)\leq x_n \rbrace).$$
The set of \emph{all} $n$-dimensional probability distributions for each $n\in\mathbb{Z}^{+}$ and each set of $t_1,\ldots,t_n\in T$ is called \emph{the} family of finite dimensional probability distributions, or family of finite
dimensional distributions, abbreviated f.f.d., of the stochastic process $\lbrace X(t)\mid t\in T\rbrace$.
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Two stochastic processes $\lbrace X(t) \mid t\in T \rbrace$ and $\lbrace Y(s) \mid s\in S \rbrace$ are said to be \emph{identically distributed}, or \emph{versions of each other} if
\begin{enumerate}
\item $S=T$, and
\item $\lbrace X(t)\rbrace$ and $\lbrace Y(s)\rbrace$ have the same f.f.d.
\end{enumerate} |
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