Kolmogorov extension theorem: Difference between revisions

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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.

The theorem is used in one of the standard proofs of existence of a [[Brownian motion]], by specifying the finite dimensional distributions to be Gaussian random variables, satisfying the consistency conditions above. As in most of the definitions of [[Brownian motion]] it is required that the sample paths are continuous almost surely, and one then uses the [[kolmogorovKolmogorov continuity theorem]] to construct a continuous modification of the process constructed by the Kolmogorov extension theorem.

==General form of the theorem==