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

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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

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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

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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

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