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 Connections
Sources
1. Kolmogorov Extension Theorem (Wikipedia)
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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.
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