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 $$\mu$$.