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 onto a finite-dimensional vector space. It can be described using a multivariate normal distribution system for any number of coordinates.

Finite-dimensional distributions of a measure

Let be a measure space. The finite-dimensional distributions of are the pushforward measures, 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 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 is related to the definition for a measure in the following way: recall that the law of is a measure on the collection of all functions from into. In general, this is an infinite-dimensional space. The finite dimensional distributions of 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 converge weakly to the corresponding finite-dimensional distributions of some probability measure, then converges weakly to.