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distributions of a stochastic process

(Definition)

Just as one can associate a random variable with its distribution , 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 individually, it says nothing about the relationships between any pair, or more generally, any finite set of random variables 's at different 's. Another way is to look at the joint probability distribution of all the random variables in a stochastic process. This way we can derive the probability distribution functions of individual random variables. However, in most stochastic processes, there are infinitely many random variables involved, and we run into trouble right away.

To resolve this, we enlarge the above set of distribution functions to include all joint probability distributions of finitely many 's, called the family of finite dimensional probability distributions. Specifically, let $n<\infty$ be any positive integer, an -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 $t_i\in T$ :

$\displaystyle F_{t_1,\ldots,t_n}(x_1,\ldots,x_n):=F_{X(t_1),\ldots,X(t_n)}(x_1,... ...lbrace X(t_1)\leq x_1 \rbrace \cap \cdots \cap \lbrace X(t_n)\leq x_n \rbrace).$

The set of all

-dimensional probability distributions for each $n\in\mathbb{Z}^{+}$ and each set of $t_1,\ldots,t_n\in T$ is called the family of finite dimensional probability distributions, or family of finite dimensional distributions, abbreviated f.f.d., of the stochastic process $\lbrace X(t)\mid t\in T\rbrace$ .

Let $\sigma$ be a permutation on $\lbrace 1,\ldots, n\rbrace$ . For any $t_1,\ldots,t_n\in T$ and $x_1,\ldots,x_n\in \mathbb{R}$ , define $s_i=t_{\sigma(i)}$ and $y_i=x_{\sigma(i)}$ . Then

$\displaystyle F_{s_1,\ldots,s_n}(y_1,\ldots,y_n)$	$\displaystyle =$	$\displaystyle P(\lbrace X(s_1)\leq y_1 \rbrace \cap \cdots \cap \lbrace X(s_n)\leq y_n \rbrace)$
	$\displaystyle =$	$\displaystyle P(\lbrace X(t_1)\leq x_1 \rbrace \cap \cdots \cap \lbrace X(t_n)\leq t_n \rbrace)$
	$\displaystyle =$	$\displaystyle F_{t_1,\ldots,t_n}(x_1,\ldots,x_n).$

We say that the finite probability distributions are consistent with one another if, for any , each set of $t_1,\ldots,t_{n+1}\in T$ ,

$\displaystyle F_{t_1,\ldots,t_n}(x_1,\ldots,x_n)=\lim_{x_{n+1}\to\infty}F_{t_1,\ldots,t_n,t_{n+1}}(x_1,\ldots,x_n,x_{n+1}).$

Two stochastic processes $\lbrace X(t) \mid t\in T \rbrace$ and $\lbrace Y(s) \mid s\in S \rbrace$ are said to be identically distributed, or versions of each other if

, and
$\lbrace X(t)\rbrace$ and $\lbrace Y(s)\rbrace$ have the same f.f.d.

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Other names:

finite dimensional probability distributions, ffd

Also defines:

finite dimensional distributions, f.f.d., identically distributed stochastic processes, version of a stochastic process

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Cross-references: identically distributed, finite, permutation, integer, positive, distribution functions, finite set, stochastic process, distribution, random variable
There are 2 references to this entry.

This is version 9 of distributions of a stochastic process, born on 2005-06-22, modified 2006-08-14.
Object id is 7183, canonical name is DistributionsOfAStochasticProcess.
Accessed 4936 times total.

Classification:

AMS MSC:

60G07 (Probability theory and stochastic processes :: Stochastic processes :: General theory of processes)

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