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)