Finite-dimensional distribution

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

Finite-dimensional distributions of a stochastic process

Let $(\mathrm{\Omega },{ℱ},\mathbb{\mathbb{P}})$ be a probability space and let $X:I\times \mathrm{\Omega }\to \mathbb{𝕏}$ be a stochastic process. The finite-dimensional distributions of $X$ are the push forward measures ${\mathbb{\mathbb{P}}}_{i_1\dots i_k}^X$ on the product space ${\mathbb{𝕏}}^k$ for $k\in \mathbb{\mathbb{N}}$ defined by

${\mathbb{\mathbb{P}}}_{i_1\dots i_k}^X(S):=\mathbb{\mathbb{P}}\{\omega \in \mathrm{\Omega }|(X_{i_1}(\omega ),\dots ,X_{i_k}(\omega ))\in S\}.$

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

${\mathbb{\mathbb{P}}}_{i_1\dots i_k}^X(A_1\times \cdots \times A_k):=\mathbb{\mathbb{P}}\{\omega \in \mathrm{\Omega }|X_{i_j}(\omega )\in A_j\mathrm{f}\mathrm{o}\mathrm{r}1\le j\le k\}.$

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 ${{ℒ}}_X$ 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 $f_{*}\left({{ℒ}}_X\right)$ on the finite-dimensional product space ${\mathbb{𝕏}}^k$, where

$f:{\mathbb{𝕏}}^I\to {\mathbb{𝕏}}^k:\sigma \mapsto (\sigma (t_1),\dots ,\sigma (t_k))$

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 $({\mu }_n{)}_{n=1}^{\mathrm{\infty }}$ 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)