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
Let [math]\displaystyle{ (X, \mathcal{F}, \mu) }[/math] be a measure space. The finite-dimensional distributions of [math]\displaystyle{ \mu }[/math] are the pushforward measures [math]\displaystyle{ f_{*} (\mu) }[/math], where [math]\displaystyle{ f : X \to \mathbb{R}^{k} }[/math], [math]\displaystyle{ k \in \mathbb{N} }[/math], is any measurable function.
Finite-dimensional distributions of a stochastic process
Let [math]\displaystyle{ (\Omega, \mathcal{F}, \mathbb{P}) }[/math] be a probability space and let [math]\displaystyle{ X : I \times \Omega \to \mathbb{X} }[/math] be a stochastic process. The finite-dimensional distributions of [math]\displaystyle{ X }[/math] are the push forward measures [math]\displaystyle{ \mathbb{P}_{i_{1} \dots i_{k}}^{X} }[/math] on the product space [math]\displaystyle{ \mathbb{X}^{k} }[/math] for [math]\displaystyle{ k \in \mathbb{N} }[/math] defined by
- [math]\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\}. }[/math]
Very often, this condition is stated in terms of measurable rectangles:
- [math]\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\}. }[/math]
The definition of the finite-dimensional distributions of a process [math]\displaystyle{ X }[/math] is related to the definition for a measure [math]\displaystyle{ \mu }[/math] in the following way: recall that the law [math]\displaystyle{ \mathcal{L}_{X} }[/math] of [math]\displaystyle{ X }[/math] is a measure on the collection [math]\displaystyle{ \mathbb{X}^{I} }[/math] of all functions from [math]\displaystyle{ I }[/math] into [math]\displaystyle{ \mathbb{X} }[/math]. In general, this is an infinite-dimensional space. The finite dimensional distributions of [math]\displaystyle{ X }[/math] are the push forward measures [math]\displaystyle{ f_{*} \left( \mathcal{L}_{X} \right) }[/math] on the finite-dimensional product space [math]\displaystyle{ \mathbb{X}^{k} }[/math], where
- [math]\displaystyle{ f : \mathbb{X}^{I} \to \mathbb{X}^{k} : \sigma \mapsto \left( \sigma (t_{1}), \dots, \sigma (t_{k}) \right) }[/math]
is the natural "evaluate at times [math]\displaystyle{ t_{1}, \dots, t_{k} }[/math]" function.
Relation to tightness
It can be shown that if a sequence of probability measures [math]\displaystyle{ (\mu_{n})_{n = 1}^{\infty} }[/math] is tight and all the finite-dimensional distributions of the [math]\displaystyle{ \mu_{n} }[/math] converge weakly to the corresponding finite-dimensional distributions of some probability measure [math]\displaystyle{ \mu }[/math], then [math]\displaystyle{ \mu_{n} }[/math] converges weakly to [math]\displaystyle{ \mu }[/math].
See also
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