Definition (Finite dimensional distribution for stochastic process)
Given a Stochastic Process {Xt;t∈T}, and k∈N, and a finite collection t1,…,tk∈T of distinct index values, we define the Borel Probability Measure μt1,…,tk on Rk bt μt1…tk(H)=P((Xt1,…,Xtk)∈H),H⊆Rk Borel.The distributions {μt1,…,tk;k∈N,t1,…,tk∈T distinct} are called the finite dimensional distributions for the stochastic process {Xt;t∈T}.
Proposition (Consistency properties of finite-dimensional distributions)
These finite-dimensional distributions satisfy two consistency conditions:
- If (s(1),s(2),…,s(k)) is any permutation of (1,2,…,k), then for distinct t1,…,tk∈T, and any Borel H1,…,Hk⊆R we have μt1…tk(H1×⋯×Hk)=μts(1)…ts(k)(Hs(1)×⋯×Hs(k)).e.g. we must have P(X∈A,Y∈B)=P(Y∈B,X∈A) but this will not usually equal P(Y∈A,X∈B).
- For distinct t1,…,tk∈T, and any Borel H1,…,Hk−1⊆R, we have μt1…tk(H1×⋯×Hk−1×R)=μt1…tk−1(H1×⋯×Hk−1).That is, allowing Xtk to be anywhere in R is equivalent to not mentioning Xtk at all. e.g. P(X∈A,Y∈R)=P(X∈A)
These conditions are quite obvious for any conceivable stochastic process. The following theorem states the converse:
Theorem (15.1.3)
A family of Borel Probability Measures {μt1…tk;k∈N,ti∈T distinct}, with μt1…tk a Measure on Rk satisfies the consistency conditions in Proposition 2 if and only if there exists a Probability Space (RT,FT,P), and Random Variables {Xt}t∈T defined on it such that ∀k∈N, and for all distinct t1,…,tk∈T, and all Borel H⊆Rk, we have P((Xt1,…,Xtk)∈H)=μt1…tk(H).
\begin{proof} The “if” direction is immediate, the “only if” direction follows if we take RT={all functions T→R} and FT=σ{{Xt∈H};t∈T,H⊆R Borel} and do some more stuff that I’m lazy to say here
\end{proof}