Kolmogorov extension theorem: Difference between revisions

Revision as of 15:24, 22 December 2015

Revision as of 13:41, 3 July 2016

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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 KolomogorovKolmogorov'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.