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 be a measure space. The finite-dimensional distributions of \mu are the pushforward measures f_{*} (\mu), where, , is any measurable function.

Finite-dimensional distributions of a stochastic process

Let be a probability space and let be a stochastic process. The finite-dimensional distributions of X are the push forward measures on the product space for defined by Very often, this condition is stated in terms of measurable rectangles: 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 of X is a measure on the collection 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 on the finite-dimensional product space, where is the natural "evaluate at times " function.

Relation to tightness

It can be shown that if a sequence of probability measures 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.