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Kolmogorov Extension Theorem

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Theorem
StochasticProcesses

Given a stochastic process {Xt;t∈T}\{ X_{t};t\in T \}, and k∈Nk\in \mathbb{N}, and a finite collection t1,…,tk∈Tt_{1},\dots,t_{k}\in T of distinct index values, we define the Borel probability measure μt1,…,tk\mu_{t_{1},\dots,t_{k}} on Rk\mathbb{R}^{k} bt μt1…tk(H)=P((Xt1,…,Xtk)∈H),H⊆Rk  Borel.\mu_{t_{1}\dots t_{k}}(H)=\mathbb{P}((X_{t_{1}},\dots,X_{t_{k}})\in H),\quad H\subseteq \mathbb{R}^{k}\,\text{ Borel.}The distributions {μt1,…,tk;k∈N,t1,…,tk∈T distinct}\{ \mu_{t_{1},\dots,t_{k}};k\in \mathbb{N},t_{1},\dots,t_{k}\in T\text{ distinct} \} are called the finite dimensional distributions for the stochastic process {Xt;t∈T}\{ X_{t};t\in T \}.

These finite-dimensional distributions satisfy two consistency conditions: 1. If (s(1),s(2),…,s(k))(s(1),s(2),\dots,s(k)) is any permutation of (1,2,…,k)(1,2,\dots,k), then for distinct t1,…,tk∈Tt_{1},\dots,t_{k}\in T, and any Borel H1,…,Hk⊆RH_{1},\dots,H_{k}\subseteq \mathbb{R} we have μt1…tk(H1×⋯×Hk)=μts(1)…ts(k)(Hs(1)×⋯×Hs(k)).\mu_{t_{1}\dots t_{k}}(H_{1}\times\dots \times H_{k})=\mu_{t_{s(1)}\dots t_{s(k)}}(H_{s(1)}\times\dots \times H_{s(k)}).e.g. we must have P(X∈A,Y∈B)=P(Y∈B,X∈A)\mathbb{P}(X\in A,Y\in B)=\mathbb{P}(Y\in B,X\in A) but this will not usually equal P(Y∈A,X∈B)\mathbb{P}(Y\in A,X\in B). 2. For distinct t1,…,tk∈Tt_{1},\dots,t_{k}\in T, and any Borel H1,…,Hk−1⊆RH_{1},\dots,H_{k-1}\subseteq \mathbb{R}, we have μt1…tk(H1×⋯×Hk−1×R)=μt1…tk−1(H1×⋯×Hk−1).\mu_{t_{1}\dots t_{k}}(H_{1}\times\dots \times H_{k-1}\times \mathbb{R})=\mu_{t_{1}\dots t_{k-1}}(H_{1}\times \dots \times H_{k-1}).That is, allowing XtkX_{t_{k}} to be anywhere in R\mathbb{R} is equivalent to not mentioning XtkX_{t_{k}} at all. e.g. P(X∈A,Y∈R)=P(X∈A)\mathbb{P}(X\in A,Y\in \mathbb{R})=\mathbb{P}(X\in A)

These conditions are quite obvious for any conceivable stochastic process. The following theorem states the converse:

A family of Borel probability measures {μt1…tk; k∈N, ti∈T distinct}\{ \mu_{t_{1}\dots t_{k}};\,k\in \mathbb{N},\,t_{i}\in T\text{ distinct} \}, with μt1…tk\mu_{t_{1}\dots t_{k}} a measure on Rk\mathbb{R}^{k} satisfies the consistency conditions in Proposition 2 if and only if there exists a probability space (RT,FT,P)(\mathbb{R}^{T},\mathcal{F}^{T},\mathbb{P}), and random variables {Xt}t∈T\{ X_{t} \}_{t\in T} defined on it such that ∀k∈N\forall k\in \mathbb{N}, and for all distinct t1,…,tk∈Tt_{1},\dots,t_{k}\in T, and all Borel H⊆RkH\subseteq \mathbb{R}^{k}, we have P((Xt1,…,Xtk)∈H)=μt1…tk(H).\mathbb{P}((X_{t_{1}},\dots,X_{t_{k}})\in H)=\mu_{t_{1}\dots t_{k}}(H).

This theorem says, that under extremely general conditions, stochastic processes exists.

\begin{proof} The “if” direction is immediate, the “only if” direction follows if we take RT={all functions T→R}\mathbb{R}^{T}=\{ \text{all functions }T\to \mathbb{R} \} and FT=σ{{Xt∈H};t∈T, H⊆R Borel}\mathcal{F}^{T}=\sigma \{ \{ X_{t}\in H \};t\in T,\,H\subseteq \mathbb{R}\text{ Borel} \} and do some more stuff that I’m lazy to say here

\end{proof}