# Finite-dimensional distribution

{{ safesubst:#invoke:Unsubst||$N=Unreferenced |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} 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 stochastic process

${\mathbb {P} }_{i_{1}\dots i_{k}}^{X}(S):={\mathbb {P} }\left\{\omega \in \Omega \left|\left(X_{i_{1}}(\omega ),\dots ,X_{i_{k}}(\omega )\right)\in S\right.\right\}.$ Very often, this condition is stated in terms of measurable rectangles:

${\mathbb {P} }_{i_{1}\dots i_{k}}^{X}(A_{1}\times \cdots \times A_{k}):={\mathbb {P} }\left\{\omega \in \Omega \left|X_{i_{j}}(\omega )\in A_{j}{\mathrm {\,for\,} }1\leq j\leq k\right.\right\}.$ $f:{\mathbb {X} }^{I}\to {\mathbb {X} }^{k}:\sigma \mapsto \left(\sigma (t_{1}),\dots ,\sigma (t_{k})\right)$ is the natural "evaluate at times $t_{1},\dots ,t_{k}$ " function.