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 $(X,{\mathcal {F}},\mu )$ 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

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

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

• Law (stochastic processes)