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).
Let ( X , F , μ ) be a measure space. The finite-dimensional distributions of μ are the pushforward measures f ∗ ( μ ) , where f : X → R k , k ∈ N , is any measurable function.
Let ( Ω , F , P ) be a probability space and let X : I × Ω → X be a stochastic process. The finite-dimensional distributions of X are the push forward measures P i 1 … i k X on the product space X k for k ∈ N defined by
P i 1 … i k X ( S ) := P { ω ∈ Ω | ( X i 1 ( ω ) , … , X i k ( ω ) ) ∈ S } . 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 } . The definition of the finite-dimensional distributions of a process X is related to the definition for a measure μ in the following way: recall that the law L X of X is a measure on the collection X I of all functions from I into X . In general, this is an infinite-dimensional space. The finite dimensional distributions of X are the push forward measures f ∗ ( L X ) on the finite-dimensional product space X k , where
f : X I → X k : σ ↦ ( σ ( t 1 ) , … , σ ( t k ) ) is the natural "evaluate at times t 1 , … , t k " function.
It can be shown that if a sequence of probability measures ( μ n ) n = 1 ∞ is tight and all the finite-dimensional distributions of the μ n converge weakly to the corresponding finite-dimensional distributions of some probability measure μ , then μ n converges weakly to μ .
