Finite-dimensional Distribution
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, finite-dimensional distributions are a tool in the study of measures and
stochastic processes In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Stoc ...
. A lot of information can be gained by studying the "projection" of a measure (or process) onto a finite-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
(or finite collection of times).


Finite-dimensional distributions of a measure

Let (X, \mathcal, \mu) be a
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
. The finite-dimensional distributions of \mu are the
pushforward measure In measure theory, a pushforward measure (also known as push forward, push-forward or image measure) is obtained by transferring ("pushing forward") a measure from one measurable space to another using a measurable function. Definition Given mea ...
s f_ (\mu), where f : X \to \mathbb^, k \in \mathbb, is any measurable function.


Finite-dimensional distributions of a stochastic process

Let (\Omega, \mathcal, \mathbb) be a
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models ...
and let X : I \times \Omega \to \mathbb be a
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
. The finite-dimensional distributions of X are the push forward measures \mathbb_^ on the
product space In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemi ...
\mathbb^ for k \in \mathbb defined by :\mathbb_^ (S) := \mathbb \left\. Very often, this condition is stated in terms of
measurable In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts hav ...
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...
s: :\mathbb_^ (A_ \times \cdots \times A_) := \mathbb \left\. The definition of the finite-dimensional distributions of a process X is related to the definition for a measure \mu in the following way: recall that the
law Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior, with its precise definition a matter of longstanding debate. It has been variously described as a science and as the ar ...
\mathcal_ of X is a measure on the collection \mathbb^ of all functions from I into \mathbb. In general, this is an infinite-dimensional space. The finite dimensional distributions of X are the push forward measures f_ \left( \mathcal_ \right) on the finite-dimensional product space \mathbb^, where :f : \mathbb^ \to \mathbb^ : \sigma \mapsto \left( \sigma (t_), \dots, \sigma (t_) \right) is the natural "evaluate at times t_, \dots, t_" function.


Relation to tightness

It can be shown that if a sequence of
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
s (\mu_)_^ is
tight Tight may refer to: Clothing * Skin-tight garment, a garment that is held to the skin by elastic tension * Tights, a type of leg coverings fabric extending from the waist to feet * Tightlacing, the practice of wearing a tightly-laced corset ...
and all the finite-dimensional distributions of the \mu_ converge weakly to the corresponding finite-dimensional distributions of some probability measure \mu, then \mu_ converges weakly to \mu.


See also

*
Law (stochastic processes) In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Stoc ...
{{DEFAULTSORT:Finite-Dimensional Distribution Measure theory Stochastic processes