Kolmogorov extension theorem

From Wikipedia, the free encyclopedia

In mathematics, the Kolmogorov extension theorem (also known as Kolmogorov existence theorem) is a theorem that guarantees that a suitably "consistent" collection of finite-dimensional distributions will define a stochastic process. It is credited to the Soviet mathematician Andrey Nikolaevich Kolmogorov.^[1]

[edit] Statement of the theorem

Let $T$ denote some interval (thought of as "time"), and let $n \in \mathbb{N}$ . For each $k \in \mathbb{N}$ and finite sequence of times $t_{1}, \dots, t_{k} \in T$ , let $\nu_{t_{1} \dots t_{k}}$ be a probability measure on $(\mathbb{R}^{n})^{k}$ . Suppose that these measures satisfy two consistency conditions:

1. for all permutations $π$ of $\{ 1, \dots, k \}$ and measurable sets $F_{i} \subseteq \mathbb{R}^{n}$ ,

$\nu_{t_{\pi (1)} \dots t_{\pi (k)}} \left( F_{1} \times \dots \times F_{k} \right) = \nu_{t_{1} \dots t_{k}} \left( F_{\pi^{-1} (1)} \times \dots \times F_{\pi^{-1} (k)} \right);$

2. for all measurable sets $F_{i} \subseteq \mathbb{R}^{n}$ , $m \in \mathbb{N}$

$\nu_{t_{1} \dots t_{k}} \left( F_{1} \times \dots \times F_{k} \right) = \nu_{t_{1} \dots t_{k} t_{k + 1}, \dots , t_{k+m}} \left( F_{1} \times \dots \times F_{k} \times \mathbb{R}^{n} \times \dots \times \mathbb{R}^{n} \right).$

Then there exists a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and a stochastic process $X : T \times \Omega \to \mathbb{R}^{n}$ such that

$\nu_{t_{1} \dots t_{k}} \left( F_{1} \times \dots \times F_{k} \right) = \mathbb{P} \left( X_{t_{1}} \in F_{1}, \dots, X_{t_{k}} \in F_{k} \right)$

for all $t_{i} \in T$ , $k \in \mathbb{N}$ and measurable sets $F_{i} \subseteq \mathbb{R}^{n}$ , i.e. $X$ has the $\nu_{t_{1} \dots t_{k}}$ as its finite-dimensional distributions.