Kolmogorov extension theorem: Difference between revisions

Revision as of 05:00, 21 June 2015

Revision as of 05:04, 21 June 2015

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Then there exists a unique probability measure <math>\mu</math> on <math>\Omega_T </math> such that <math>\mu_F=(\pi_{F \leftarrow T})_* \mu</math> for every finite subset <math>F \subset T</math>.

Note that the original statement of the theorem is just a special case of this theorem with <math>\Omega_t = \mathbb{R}^n </math> for all <math>t \in T</math>, and <math> \mu_{\{t_1,...,t_k\}}=\nu_{t_1 \dots t_k}</math>. 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 existence of[[Wiener process]],
*a [[Markov chain]] taking values in a given (discrete) state space with a given transition matrix,
*the existence of random-cluster modelsmodel on infinite lattices with given parameters <math>p,q</math>, as well as the existence of
*infinite products of (inner-regular) probability spaces.

==References==