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{{Short description|Finite-dimensional distribution}}
In [[Mathematics|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|vector space]] (or finite collection of times).

==Finite-dimensional distributions of a measure==
Let <math>(X, \mathcal{F}, \mu)</math> be a [[Measure space|measure space]]. The '''finite-dimensional distributions''' of <math>\mu</math> are the [[Pushforward measure|pushforward measure]]s <math>f_{*} (\mu)</math>, where <math>f : X \to \mathbb{R}^{k}</math>, <math>k \in \mathbb{N}</math>, is any measurable function.

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

Very often, this condition is stated in terms of measurable [[Rectangle|rectangle]]s:
:<math>\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\}.</math>

The definition of the finite-dimensional distributions of a process <math>X</math> is related to the definition for a measure <math>\mu</math> in the following way: recall that the [[Law (stochastic processes)|law]] <math>\mathcal{L}_{X}</math> of <math>X</math> is a measure on the collection <math>\mathbb{X}^{I}</math> of all functions from <math>I</math> into <math>\mathbb{X}</math>. In general, this is an infinite-dimensional space. The finite dimensional distributions of <math>X</math> are the push forward measures <math>f_{*} \left( \mathcal{L}_{X} \right)</math> on the finite-dimensional product space <math>\mathbb{X}^{k}</math>, where
:<math>f : \mathbb{X}^{I} \to \mathbb{X}^{k} : \sigma \mapsto \left( \sigma (t_{1}), \dots, \sigma (t_{k}) \right)</math>
is the natural "evaluate at times <math>t_{1}, \dots, t_{k}</math>" function.

==Relation to tightness==
It can be shown that if a sequence of [[Probability measure|probability measure]]s <math>(\mu_{n})_{n = 1}^{\infty}</math> is [[Tightness of measures|tight]] and all the finite-dimensional distributions of the <math>\mu_{n}</math> converge weakly to the corresponding finite-dimensional distributions of some probability measure <math>\mu</math>, then <math>\mu_{n}</math> converges weakly to <math>\mu</math>.

==See also==
* [[Law (stochastic processes)]]

{{DEFAULTSORT:Finite-Dimensional Distribution}}
[[Category:Measure theory]]
[[Category:Stochastic processes]]

{{Sourceattribution|Finite-dimensional distribution}}