Kolmogorov extension theorem: Difference between revisions

Revision as of 04:56, 29 July 2015

Revision as of 21:25, 8 August 2015

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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 <math>\nu_{t_{1} \dots t_{k}}</math> as its finite-dimensional distributions relative to times <math>t_{1} \dots t_{k}</math>.

In fact, it is always possible to take as the underlying probability space <math>\Omega = (\mathbb{R}^n)^T</math> and to take for <math>X</math> the canonical process <math>X\colon (t,Y) \mapsto Y_t</math>. Therefore, an alternative way of stating Kolomogorov's extension theorem is that, provided that the above consistency conditions hold, there exists a (unique) measure <math>\nu</math> on <math>(\mathbb{R}^n)^T</math> with marginals <math>\nu_{t_{1} \dots t_{k}}</math> for any finite collection of times <math>t_{1} \dots t_{k}</math>. Kolmogorov's extension theorem applies when <math>T</math> is uncountable, but the price to pay

for this level of generality is that the measure <math>\nu</math> is only defined on the product [[σ-algebra]] of <math>(\mathbb{R}^n)^T</math>, which is not very rich.

==Explanation of the conditions==

The two conditions required by the theorem are trivially satisfied by any stochastic process. For example, consider a real-valued discrete-time stochastic process <math>X</math>. Then the probability <math>\mathbb{P}(X_1 >0, X_2<0)</math> can be computed either as <math>\nu_{1,2}( \mathbb{R}_+ \times \mathbb{R}_-)</math> or as <math>\nu_{2,1}( \mathbb{R}_- \times \mathbb{R}_+)</math>. Hence, for the finite-dimensional distributions to be consistent, it must hold that

<math>\nu_{1,2}( \mathbb{R}_+ \times \mathbb{R}_-) = \nu_{2,1}( \mathbb{R}_- \times \mathbb{R}_+)</math>.

The first condition generalises this obvious statement to hold for any number of time points <math>t_i</math>, and any control sets <math>F_i</math>.

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==A more general form of the theorem==

The Kolmogorov extension theorem gives us conditions for a collection of measures on Euclidean spaces to be the finite-dimensional distributions of some <math>\mathbb{R}^{n}</math>-valued stochastic process, but the assumption that the state space be <math>\mathbb{R}^{n}</math> is unnecessary. In fact, any collection of measurable spaces together with a collection of [[inner regular measure]]s defined on the finite products of these spaces would suffice, provided that these measures satisfy a certain compatibility relation. The formal statement of the general theorem is as follows.<ref>T. Tao, ''An Introduction to Measure Theory'', [[Graduate Studies in Mathematics]], Vol. 126, 2011, p. 195</ref>

Let <math>T</math> be any set. Let <math> \{ (\Omega_t, \mathcal{F}_t) \}_{t \in T} </math> be some collection of measurable spaces, and for each <math> t \in T </math>, let <math> \tau_t</math> be a Hausdorff topology on <math> \Omega_t</math>. For each subset <math>J \subset T</math>, define