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Finite-dimensional distributions of a measure

Let ( X , F , μ ) {\displaystyle (X,{\mathcal {F}},\mu )} be a measure space. The finite-dimensional distributions of μ {\displaystyle \mu } are the pushforward measures f ∗ ( μ ) {\displaystyle f_{*}(\mu )} , where f : X → R k {\displaystyle f:X\to \mathbb {R} ^{k}} , k ∈ N {\displaystyle k\in \mathbb {N} } , is any measurable function.

Finite-dimensional distributions of a stochastic process

Let ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )} be a probability space and let X : I × Ω → X {\displaystyle X:I\times \Omega \to \mathbb {X} } be a stochastic process. The finite-dimensional distributions of X {\displaystyle X} are the push forward measures P i 1 … i k X {\displaystyle \mathbb {P} _{i_{1}\dots i_{k}}^{X}} on the product space X k {\displaystyle \mathbb {X} ^{k}} for k ∈ N {\displaystyle k\in \mathbb {N} } defined by

P i 1 … i k X ( S ) := P { ω ∈ Ω | ( X i 1 ( ω ) , … , X i k ( ω ) ) ∈ S } . {\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\}.}

Very often, this condition is stated in terms of measurable rectangles:

P i 1 … i k X ( A 1 × ⋯ × A k ) := P { ω ∈ Ω | X i j ( ω ) ∈ A j f o r 1 ≤ j ≤ k } . {\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\}.}

The definition of the finite-dimensional distributions of a process X {\displaystyle X} is related to the definition for a measure μ {\displaystyle \mu } in the following way: recall that the law L X {\displaystyle {\mathcal {L}}_{X}} of X {\displaystyle X} is a measure on the collection X I {\displaystyle \mathbb {X} ^{I}} of all functions from I {\displaystyle I} into X {\displaystyle \mathbb {X} } . In general, this is an infinite-dimensional space. The finite dimensional distributions of X {\displaystyle X} are the push forward measures f ∗ ( L X ) {\displaystyle f_{*}\left({\mathcal {L}}_{X}\right)} on the finite-dimensional product space X k {\displaystyle \mathbb {X} ^{k}} , where

f : X I → X k : σ ↦ ( σ ( t 1 ) , … , σ ( t k ) ) {\displaystyle f:\mathbb {X} ^{I}\to \mathbb {X} ^{k}:\sigma \mapsto \left(\sigma (t_{1}),\dots ,\sigma (t_{k})\right)}

is the natural "evaluate at times t 1 , … , t k {\displaystyle t_{1},\dots ,t_{k}} " function.

Relation to tightness

It can be shown that if a sequence of probability measures ( μ n ) n = 1 ∞ {\displaystyle (\mu _{n})_{n=1}^{\infty }} is tight and all the finite-dimensional distributions of the μ n {\displaystyle \mu _{n}} converge weakly to the corresponding finite-dimensional distributions of some probability measure μ {\displaystyle \mu } , then μ n {\displaystyle \mu _{n}} converges weakly to μ {\displaystyle \mu } .

See also

• Law (stochastic processes)