Kolmogorov extension theorem

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]

Statement of the theorem

Let <math>T</math> denote some interval (thought of as "time"), and let <math>n \in \mathbb{N}</math>. For each <math>k \in \mathbb{N}</math> and finite sequence of times <math>t_{1}, \dots, t_{k} \in T</math>, let <math>\nu_{t_{1} \dots t_{k}}</math> be a probability measure on <math>(\mathbb{R}^{n})^{k}</math>. Suppose that these measures satisfy two consistency conditions: 1. for all permutations <math>\pi</math> of <math>\{ 1, \dots, k \}</math> and measurable sets <math>F_{i} \subseteq \mathbb{R}^{n}</math>,

<math>\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);</math>

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

<math>\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).</math>

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

<math>\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)</math>

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

References