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

Finite-dimensional distributions of a stochastic process

Let $(\Omega, \mathcal, \mathbb)$ be a probability space and let $X : I \times \Omega \to \mathbb$ be a stochastic process. The finite-dimensional distributions of $X$ are the push forward measures $\mathbb_^$ on the product space $\mathbb^$ for $k \in \mathbb$ defined by
:$\mathbb_^ (S) := \mathbb \left\.$

Very often, this condition is stated in terms of measurable rectangles:
:$\mathbb_^ (A_ \times \cdots \times A_) := \mathbb \left\.$

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 $\mathcal_$ of $X$ is a measure on the collection $\mathbb^$ of all functions from $I$ into $\mathbb$. In general, this is an infinite-dimensional space. The finite dimensional distributions of $X$ are the push forward measures $f_ \left( \mathcal_ \right)$ on the finite-dimensional product space $\mathbb^$, where
:$f : \mathbb^ \to \mathbb^ : \sigma \mapsto \left( \sigma (t_), \dots, \sigma (t_) \right)$
is the natural "evaluate at times $t_, \dots, t_$" function.

Relation to tightness

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

See also

* Law (stochastic processes)

{{DEFAULTSORT:Finite-Dimensional Distribution
Measure theory
Stochastic processes