Ergodic theorem

An ergodic theorem relates time averages along trajectories of a measure-preserving dynamical system to spatial averages with respect to an invariant measure.

Birkhoff ergodic theorem: A theorem stating that time averages of an integrable function converge almost everywhere to an invariant function. It gives the central pointwise convergence result for discrete measure-preserving systems.
Dynamical system: A space together with a rule describing how its points evolve over time. An ergodic theorem studies averages generated by repeated evolution.
Ludwig Boltzmann: An Austrian physicist who developed statistical mechanics and the kinetic theory of gases. His ergodic hypothesis motivated mathematical efforts to justify statistical averages.
Strong law of large numbers: A probability theorem stating that sample averages converge almost surely to an expected value under suitable conditions. It can be viewed as an ergodic theorem for independent identically distributed sequences.
Mixing (mathematics): A dynamical property in which correlations between events decay as their time separation grows. Mixing is stronger than ergodicity and yields finer information than average convergence alone.
von Neumann mean ergodic theorem: A theorem asserting convergence of averaged iterates in a Hilbert space for a unitary operator. It establishes convergence in mean, complementing Birkhoff’s almost-everywhere convergence.
Time average: An average formed from values observed along a system’s trajectory over time. It is the trajectory-based quantity whose convergence the theorem describes.
George David Birkhoff: An American mathematician known for foundational work in dynamical systems and ergodic theory. His 1931 theorem established a central pointwise result for time averages.
Birkhoff–Khinchin theorem: A pointwise ergodic theorem for stationary stochastic processes. It translates ergodic averaging into convergence of averages along stationary sequences.
Uniform convergence: Convergence in which one index threshold works uniformly over every point in a domain. Ergodic theorems generally provide almost-everywhere or norm convergence, not uniform convergence.

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Lochs's theoremRelated: Ergodic averages provide the typical growth rates underlying continued-fraction statistics.