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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
Finite-dimensional distributions of a stochastic process
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 .