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 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 $X(t_1),\mathrm{\dots },X(t_n)$, where $t_i\in T$:

$$F_{t_1,\mathrm{\dots },t_n}(x_1,\mathrm{\dots },x_n):=F_{X(t_1),\mathrm{\dots },X(t_n)}(x_1,\mathrm{\dots },x_n)=P(\{X(t_1)\le x_1\}\cap \mathrm{\cdots }\cap \{X(t_n)\le x_n\}).$$

The set of all $n$-dimensional probability distributions for each $n\in {\mathbb{Z}}^{+}$ and each set of $t_1,\mathrm{\dots },t_n\in T$ 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 $\{1,\mathrm{\dots },n\}$. For any $t_1,\mathrm{\dots },t_n\in T$ and $x_1,\mathrm{\dots },x_n\in ℝ$, define $s_i=t_{\sigma (i)}$ and $y_i=x_{\sigma (i)}$. Then

$F_{s_1,\mathrm{\dots },s_n}(y_1,\mathrm{\dots },y_n)$ $=$ $P(\{X(s_1)\le y_1\}\cap \mathrm{\cdots }\cap \{X(s_n)\le y_n\})$

$=$ $P(\{X(t_1)\le x_1\}\cap \mathrm{\cdots }\cap \{X(t_n)\le x_n\})$

$=$ $F_{t_1,\mathrm{\dots },t_n}(x_1,\mathrm{\dots },x_n).$

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

$$F_{t_1,\mathrm{\dots },t_n}(x_1,\mathrm{\dots },x_n)=\underset{x_{n+1}\to \mathrm{\infty }}{lim}{F}_{t_1,\mathrm{\dots },t_n,t_{n+1}}(x_1,\mathrm{\dots },x_n,x_{n+1}).$$

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. $S=T$, and

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