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Birkhoff’s theorem (see Birkhoff 1931) extends the strong law of large numbers to stationary processes. The theorem is most easily formulated in terms of measure-preserving transformations: If \((\Omega ,\mathcal {F},P )\) is a probability space then a measurable transformation \(T:\Omega \rightarrow \Omega \) is measure-preserving if EX = EX ∘ T for every bounded random variable X defined on Ω. In this case P is said to be T − invariant.

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Ergodic Theorem, The

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  • 发表日期：

    2025年

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  • Steven P. Lalley

摘要

Birkhoff’s theorem (see Birkhoff 1931) extends the strong law of large numbers to stationary processes. The theorem is most easily formulated in terms of measure-preserving transformations: If \((\Omega ,\mathcal {F},P )\) is a probability space then a measurable transformation \(T:\Omega \rightarrow \Omega \) is measure-preserving if EX = EX ∘ T for every bounded random variable X defined on Ω. In this case P is said to be T − invariant.