Kolmogorov extension theorem: Difference between revisions

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In [[mathematics]], the '''Kolmogorov extension theorem''' or '''Daniell-Kolmogorov extension theorem''' (also known as '''Kolmogorov existence theorem''' or '''Kolmogorov consistency 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 [[Soviet Union|Soviet]] [[mathematician]] [[Andrey Kolmogorov|Andrey Nikolaevich Kolmogorov]]<ref>{{cite book | author=Øksendal, Bernt | title=Stochastic Differential Equations: An Introduction with Applications | publisher=Springer, Berlin | year=2003 | isbn=3-540-04758-1}}</ref> and also to [[United Kingdom|British]] mathematician [[Percy John Daniell]] who discovered it independently in the slightly different setting of integration theory. <ref>J. Aldrich, But you have to remember PJ Daniell of Sheffield, Electronic Journal for History of Probability and Statistics, Vol. 3, number 2, 2007 </ref>

==Statement of the theorem==

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==Implications of the theorem==

Since the two conditions are trivially satisfied for any stochastic process, the power of the theorem is that no other conditions are required: For any reasonable (i.e., consistent) family of finite-dimensional distributions, there exists a stochastic process with these distributions.

The measure-theoretic approach to stochastic processes starts with a probability space and defines a stochastic process as a family of functions on this probability space. However, in many applications the starting point is really the finite-dimensional distributions of the stochastic process. The theorem says that provided the finite-dimensional distributions satisfy the obvious consistency requirements, one can always identify a probability space to match the purpose. In many situations, this means that one does not have to be explicit about what the probability space is. Many texts on stochastic processes do, indeed, assume a probability space but never state explicitly what it is.

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==A more general form of the theorem==

The Kolmogorov extentionextension 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

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For subsets <math>I \subset J \subset T</math>, let <math>\pi_{I \leftarrow J}: \Omega_J \to \Omega_I</math> denote the canonical projection map <math> \omega \mapsto \omega|_I </math>.

For each finite subset <math> F \subset T</math>, suppose we have a probability measure <math> \mu_F </math> on <math> \Omega_F </math> which is [[inner regular]] with respect to the product topology (induced by the <math>\tau_t</math>) on <math>\Omega_F </math>. Suppose also that this collection <math>\{\mu_F\}</math> of measures satisfies the following compatibility relation: for finite subsets <math>F \subset G \subset T</math>, we have that

:<math>\mu_F = (\pi_{F \leftarrow G})_* \mu_G</math>

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This theorem has many far-reaching consequences; for example it can be used to prove the existence of the following, among others:

*Brownian motion, i.e., the [[Wiener process]],

*a [[Markov chain]] taking values in a given state space with a given transition matrix,

*the random-cluster model on infinite lattices with given parameters <math>p,q</math>,

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==External links==

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:Stochastic processes]]