# Ergodic Theorems

• Robert M. Gray

## Abstract

At the heart of ergodic theory are the ergodic theorems: results providing sufficient conditions for dynamical systems or random processes to possess ergodic properties, that is, for sample averages of the form
$$< f > _n = \tfrac{1}{n}\sum\limits_{i = 0}^{n - 1} {fT^i }$$
to converge to an invariant limit. Traditional developments of the pointwise ergodic theorem focus on stationary systems and use a subsidiary result known as the maximal ergodic lemma (or theorem) to prove the ergodic theorem. The general result for AMS systems then follows since an AMS source inherits ergodic properties from its stationary mean; that is, since the set {x: < f > n (x) converges } is invariant and since a system and its stationary mean place equal probability on all invariant sets, one will possess almost everywhere ergodic properties with respect to a class of measurements if and only if the other one does and the limiting sample averages will be the same.

## Keywords

Ergodic Theorem Invariant Function Prob Ability Ergodic Measure Ergodic Property
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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