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
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
![{\displaystyle \mathbb {P} _{i_{1}\dots i_{k}}^{X}(S):=\mathbb {P} \left\{\omega \in \Omega \left|\left(X_{i_{1}}(\omega ),\dots ,X_{i_{k}}(\omega )\right)\in S\right.\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e5af19d1b87e1a6511cd2a5d1132cf983257327)
Very often, this condition is stated in terms of measurable rectangles:
![{\displaystyle \mathbb {P} _{i_{1}\dots i_{k}}^{X}(A_{1}\times \cdots \times A_{k}):=\mathbb {P} \left\{\omega \in \Omega \left|X_{i_{j}}(\omega )\in A_{j}\mathrm {\,for\,} 1\leq j\leq k\right.\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d63f816309242edc1609da207865f544f9b6b482)
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
![{\displaystyle f:\mathbb {X} ^{I}\to \mathbb {X} ^{k}:\sigma \mapsto \left(\sigma (t_{1}),\dots ,\sigma (t_{k})\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39274a48e9827835d68e89d7bdf57482fb8f341d)
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
.
See also