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 $f\; :\; X\; o\; mathbb\{R\}^\{k\}$, $k\; in\; mathbb\{N\}$, is any measurable function.

Finite-dimensional distributions of a stochastic process

Let $(Omega,\; mathcal\{F\},\; mathbb\{P\})$ be a probability space and let $X\; :\; I\; imes\; Omega\; o\; mathbb\{X\}$ 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:$mathbb\{P\}\_\{i\_\{1\}\; dots\; i\_\{k^\{X\}\; (S)\; :=\; mathbb\{P\}\; left\{\; omega\; in\; Omega\; left|\; left(\; X\_\{i\_\{1\; (omega),\; dots,\; X\_\{i\_\{k\; (omega)\; ight)\; in\; S\; ight.\; ight\}.$

Very often, this condition is stated in terms of measurable rectangles::$mathbb\{P\}\_\{i\_\{1\}\; dots\; i\_\{k^\{X\}\; (A\_\{1\}\; imes\; cdots\; imes\; A\_\{k\})\; :=\; mathbb\{P\}\; left\{\; omega\; in\; Omega\; left|\; X\_\{i\_\{j\; (omega)\; in\; A\_\{j\}\; mathrm\{,for,\}\; 1\; leq\; j\; leq\; k\; ight.\; ight\}.$

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 $f\_\{*\}\; left(\; mathcal\{L\}\_\{X\}\; ight)$ on the finite-dimensional product space $mathbb\{X\}^\{k\}$, where:$f\; :\; mathbb\{X\}^\{I\}\; o\; mathbb\{X\}^\{k\}\; :\; sigma\; mapsto\; left(\; sigma\; (t\_\{1\}),\; dots,\; sigma\; (t\_\{k\})\; ight)$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 $mu$.

ee also

* Law (stochastic processes)