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). It can be described using a multivariate normal distribution system for any number of coordinates.

Finite-dimensional distributions of a measure

be a measure space. The finite-dimensional distributions are the pushforward measures, is any measurable function.

Finite-dimensional distributions of a stochastic process

be a probability space be a stochastic process. The finite-dimensional distributions on the product space defined by

Very often, this condition is stated in terms of measurable rectangles:

in the following way: recall that the law, where

" function.

Relation to tightness

It can be shown that if a sequence of probability measures is tight converge weakly.

References

See also

• Law (stochastic processes)