Finite-dimensional distribution explained

In mathematics, finite-dimensional distributions are a tool in the study of measures and stochastic processes. A lot of information can be gained by studying the "projection" of a measure (or process) onto a finite-dimensional vector space (or finite collection of times).

Finite-dimensional distributions of a measure

Let

(X, l{F}, \mu) be a measure space. The finite-dimensional distributions of \mu are the pushforward measures f _{*} (\mu) , where f : X \to R ^{k} , k \in N , is any measurable function.

Finite-dimensional distributions of a stochastic process

Let

(\Omega, l{F}, P ) be a probability space and let X : I x \Omega \to X be a stochastic process. The finite-dimensional distributions of X are the push forward measures P ^{X} _{i _{1} ... i _{k}} on the product space X ^{k} for k \in N defined byP ^{X} _{i _{1} ... i _{k}} (S) : = P \left\{ \omega \in \Omega \left| \left( X _{i _{1}} (\omega), ... , X _{i _{k}} (\omega) \right) \in S \right. \right\}.

Very often, this condition is stated in terms of measurable rectangles:

P ^{X} _{i _{1} ... i _{k}} (A _{1} x … x A _{k} ) : = P \left\{ \omega \in \Omega \left| X _{i _{j}} (\omega) \in A _{j} for 1 \leq j \leq k \right. \right\}.

The definition of the finite-dimensional distributions of a process

X is related to the definition for a measure \mu in the following way: recall that the law l{L} _{X} of X is a measure on the collection X ^{I} of all functions from I into X . In general, this is an infinite-dimensional space. The finite dimensional distributions of X are the push forward measures f _{*} \left( l{L} _{X} \right) on the finite-dimensional product space X ^{k} , wheref : X ^{I} \to X ^{k} : \sigma \mapsto \left( \sigma (t _{1} ), ... , \sigma (t _{k} ) \right) is the natural "evaluate at times t _{1} , ... , t _{k} " function.

Relation to tightness

It can be shown that if a sequence of probability measures

(\mu _{n} ) ^{inf ty} _{n =1} is tight and all the finite-dimensional distributions of the \mu _{n} converge weakly to the corresponding finite-dimensional distributions of some probability measure \mu , then \mu _{n} converges weakly to \mu .

See also

• Law (stochastic processes)