distributions of a stochastic process

Just as one can associate a random variableMathworldPlanetmath X with its distributionPlanetmathPlanetmathPlanetmath FX, one can associate a stochastic processMathworldPlanetmath {X(t)tT} with some distributions, such that the distributions will more or less describe the process. While the set of distributions {FX(t)tT} can describe the random variables X(t) individually, it says nothing about the relationships between any pair, or more generally, any finite setMathworldPlanetmath 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< be any positive integer, an n-dimensional probability distribution of the stochastic process {X(t)tT} is a joint probability distribution of X(t1),,X(tn), where tiT:


The set of all n-dimensional probability distributions for each n+ and each set of t1,,tnT 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)tT}.

Let σ be a permutation on {1,,n}. For any t1,,tnT and x1,,xn, define si=tσ(i) and yi=xσ(i). Then

Fs1,,sn(y1,,yn) = P({X(s1)y1}{X(sn)yn})
= P({X(t1)x1}{X(tn)xn})
= Ft1,,tn(x1,,xn).

We say that the finite probability distributions are consistent with one another if, for any n, each set of t1,,tn+1T,


Two stochastic processes {X(t)tT} and {Y(s)sS} 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