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.
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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
Statement Form Kolmogorov Extension Theorem
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Source Kolmogorov Extension Theorem (Wikipedia)
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Source Kolmogorov Extension Theorem (Wikipedia)
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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.
Principal Result Of: Andrey Kolmogorov, Lead sentence of the article on the result
It is credited to the English mathematician Percy John Daniell and the Russian mathematician Andrey Nikolaevich Kolmogorov.
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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