Finite-dimensional distribution

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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[edit]

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

Finite-dimensional distributions of a stochastic process[edit]

Let (\Omega, \mathcal{F}, \mathbb{P}) be a probability space and let X : I \times \Omega \to \mathbb{X} be a stochastic process. The finite-dimensional distributions of X are the push forward measures \mathbb{P}_{i_{1} \dots i_{k}}^{X} on the product space \mathbb{X}^{k} for k \in \mathbb{N} defined by

\mathbb{P}_{i_{1} \dots i_{k}}^{X} (S) := \mathbb{P} \left\{ \omega \in \Omega \left| \left( X_{i_{1}} (\omega), \dots, X_{i_{k}} (\omega) \right) \in S \right. \right\}.

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

\mathbb{P}_{i_{1} \dots i_{k}}^{X} (A_{1} \times \cdots \times A_{k}) := \mathbb{P} \left\{ \omega \in \Omega \left| X_{i_{j}} (\omega) \in A_{j} \mathrm{\,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 \mathcal{L}_{X} of X is a measure on the collection \mathbb{X}^{I} of all functions from I into \mathbb{X}. In general, this is an infinite-dimensional space. The finite dimensional distributions of X are the push forward measures f_{*} \left( \mathcal{L}_{X} \right) on the finite-dimensional product space \mathbb{X}^{k}, where

f : \mathbb{X}^{I} \to \mathbb{X}^{k} : \sigma \mapsto \left( \sigma (t_{1}), \dots, \sigma (t_{k}) \right)

is the natural "evaluate at times t_{1}, \dots, t_{k}" function.

Relation to tightness[edit]

It can be shown that if a sequence of probability measures (\mu_{n})_{n = 1}^{\infty} 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[edit]