Kolmogorov extension theorem: Difference between revisions

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{{short description|Consistent set of finite-dimensional distributions will define a stochastic process}}

{{About|a theorem deals withon stochastic processes|a theorem deals withon extension of pre-measure|Hahn–Kolmogorov theorem}}

In [[mathematics]], the '''Kolmogorov extension theorem''' (also known as '''Kolmogorov existence theorem''', the '''Kolmogorov consistency theorem''' or the '''Daniell-Kolmogorov theorem''') is a [[theorem]] that guarantees that a suitably "consistent" collection of [[finite-dimensional distribution]]s will define a [[stochastic process]]. It is credited to the English mathematician [[Percy John Daniell]] and the [[Russia|Russian]] [[mathematician]] [[Andrey Kolmogorov|Andrey Nikolaevich Kolmogorov]].<ref>{{cite book | author=Øksendal, Bernt | title=Stochastic Differential Equations: An Introduction with Applications | publisher=Springer |___location=Berlin | year=2003 |edition=Sixth | isbn=3-540-04758-1 |page=11 |url=https://books.google.com/books?id=VgQDWyihxKYC&pg=PA11 }}</ref>.

==Statement of the theorem==

Let <math>T</math> denote some [[Interval (mathematics)|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 distinct 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 [[permutation]]s <math>\pi</math> of <math>\{ 1, \dots, k \}</math> and measurable sets <math>F_{i} \subseteq \mathbb{R}^{n}</math>,

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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 Kolmogorov'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#Product_σ-algebra|product σ-algebra]] of <math>(\mathbb{R}^n)^T</math>, which is not very rich.

==Explanation of the conditions==

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* Aldrich, J. (2007) [http://www.emis.de/journals/JEHPS/Decembre2007/Aldrich.pdf "But you have to remember P.J.Daniell of Sheffield"] [http://www.emis.de/journals/JEHPS/indexang.html Electronic Journ@l for History of Probability and Statistics] December 2007.

[[Category:Theorems regardingabout stochastic processes]]