# distributions of a stochastic process

Just as one can associate a random variable $X$ with its distribution $F_{X}$, one can associate a stochastic process $\{X(t)\mid t\in T\}$ with some distributions, such that the distributions will more or less describe the process. While the set of distributions $\{F_{X(t)}\mid t\in T\}$ 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. 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 $X(t)$’s, called the family of finite dimensional probability distributions. Specifically, let $n<\infty$ be any positive integer, an $n$-dimensional probability distribution of the stochastic process $\{X(t)\mid t\in T\}$ is a joint probability distribution of $X(t_{1}),\ldots,X(t_{n})$, where $t_{i}\in T$:

 $F_{t_{1},\ldots,t_{n}}(x_{1},\ldots,x_{n}):=F_{X(t_{1}),\ldots,X(t_{n})}(x_{1}% ,\ldots,x_{n})=P(\{X(t_{1})\leq x_{1}\}\cap\cdots\cap\{X(t_{n})\leq x_{n}\}).$

The set of all $n$-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 $\{X(t)\mid t\in T\}$.

Let $\sigma$ be a permutation on $\{1,\ldots,n\}$. 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

 $\displaystyle F_{s_{1},\ldots,s_{n}}(y_{1},\ldots,y_{n})$ $\displaystyle=$ $\displaystyle P(\{X(s_{1})\leq y_{1}\}\cap\cdots\cap\{X(s_{n})\leq y_{n}\})$ $\displaystyle=$ $\displaystyle P(\{X(t_{1})\leq x_{1}\}\cap\cdots\cap\{X(t_{n})\leq x_{n}\})$ $\displaystyle=$ $\displaystyle F_{t_{1},\ldots,t_{n}}(x_{1},\ldots,x_{n}).$

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

 $F_{t_{1},\ldots,t_{n}}(x_{1},\ldots,x_{n})=\lim_{x_{n+1}\to\infty}F_{t_{1},% \ldots,t_{n},t_{n+1}}(x_{1},\ldots,x_{n},x_{n+1}).$

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

1. 1.

$S=T$, and

2. 2.

$\{X(t)\}$ and $\{Y(s)\}$ have the same f.f.d.

 Title distributions of a stochastic process Canonical name DistributionsOfAStochasticProcess Date of creation 2013-03-22 15:21:35 Last modified on 2013-03-22 15:21:35 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 13 Author CWoo (3771) Entry type Definition Classification msc 60G07 Synonym finite dimensional probability distributions Synonym ffd Related topic StochasticProcess Related topic KolmogorovsContinuityTheorem Related topic ModificationOfAStochasticProcess Defines finite dimensional distributions Defines f.f.d. Defines identically distributed stochastic processes Defines version of a stochastic process