The Kolmogorov Extension Theorem gives conditions under which a consistent family of finite-dimensional probability distributions can be extended to a single probability measure on an infinite-dimensional space of sequences or functions. Named for Andrey Kolmogorov, it is the foundational result that justifies constructing stochastic processes, such as Brownian motion, from their finite-dimensional distributions alone.
Facts
StatementThe Kolmogorov Extension Theorem states that given a suitably consistent family of finite-dimensional probability distributions, one for every finite subset of an index set, there exists a stochastic process on the whole index set whose finite-dimensional distributions are exactly that family. 1 Classification
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Source Kolmogorov Extension Theorem (Wikipedia)
Sources
1. Kolmogorov Extension Theorem (Wikipedia)
Wikimedia FoundationLead section, opening sentence
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 distributions will define a stochastic process.
Proved By: Andrey Kolmogorov, Lead paragraph
It is credited to the English mathematician Percy John Daniell and the Russian mathematician Andrey Nikolaevich Kolmogorov.
In Group: Stochastic Processes and Martingales, Lead section
of finite-dimensional distributions will define a stochastic process. It is credited to the English mathematician
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