Finite-dimensional distribution

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

Finite-dimensional distributions of a stochastic process

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

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

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

${\displaystyle \mathbb {P} _{i_{1}\dots i_{k}}^{X}(A_{1}\times \cdots \times A_{k}):=\mathbb {P} \left\{\omega \in \Omega \left|X_{i_{j}}(\omega )\in A_{j}\mathrm {\,for\,} 1\leq j\leq k\right.\right\}.}$

The definition of the finite-dimensional distributions of a process ${\displaystyle X}$ is related to the definition for a measure ${\displaystyle \mu }$ in the following way: recall that the law ${\displaystyle {\mathcal {L}}_{X}}$ of ${\displaystyle X}$ is a measure on the collection ${\displaystyle \mathbb {X} ^{I}}$ of all functions from ${\displaystyle I}$ into ${\displaystyle \mathbb {X} }$. In general, this is an infinite-dimensional space. The finite dimensional distributions of ${\displaystyle X}$ are the push forward measures ${\displaystyle f_{*}\left({\mathcal {L}}_{X}\right)}$ on the finite-dimensional product space ${\displaystyle \mathbb {X} ^{k}}$, where

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

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

Relation to tightness

It can be shown that if a sequence of probability measures ${\displaystyle (\mu _{n})_{n=1}^{\infty }}$ is tight and all the finite-dimensional distributions of the ${\displaystyle \mu _{n}}$ converge weakly to the corresponding finite-dimensional distributions of some probability measure ${\displaystyle \mu }$, then ${\displaystyle \mu _{n}}$ converges weakly to ${\displaystyle \mu }$.