Finite-dimensional distribution

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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[edit]

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[edit]

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[edit]

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 .

See also[edit]