Kolmogorov’s extension theorem

For all t1,,tk, k, let vt1,,tk be probability measuresMathworldPlanetmath on nk satisfying the following properties (consistency conditions):

  1. 1.

    vtσ(1),,tσ(k)(F1××Fk)=vt1,tk(Fσ-1(1)×Fσ-1(k)) for all permutationsMathworldPlanetmath σ of {1,2,,k} and for all Borel sets Fi of n

  2. 2.

    vt1,,tk(F1××Fk)=vt1,,tk,tk+1,tk+m(F1××Fk×n××n) for all m and for all Borel sets Fi of n

Then there exists a probability space (Ω,,P) and a stochastic processMathworldPlanetmath Xt on Ω, indexed by T, taking values in n such that


for all tiT,kn and all Borel sets Fi of n

Title Kolmogorov’s extension theorem
Canonical name KolmogorovsExtensionTheorem
Date of creation 2013-04-12 21:33:32
Last modified on 2013-04-12 21:33:32
Owner Filipe (28191)
Last modified by Filipe (28191)
Numerical id 3
Author Filipe (28191)
Entry type Theorem
Classification msc 60G07