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).
The finite-dimensional distributions of
are the pushforward measures
, is any measurable function.
be a probability space and let
be a stochastic process.
The finite-dimensional distributions of
are the push forward measures
on the product space
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
is a measure on the collection
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
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
converges weakly to