A Simple Non-Stationary Mean Ergodic Theorem, with Bonus Weak Law of Large Numbers

TL;DR

This paper provides a clear proof of a non-stationary mean ergodic theorem, showing convergence of time averages to the mean under a covariance growth condition, and includes a weak law of large numbers.

Contribution

It offers a pedagogical re-proof of a known but scattered result, clarifying conditions for convergence in non-stationary time series.

Findings

Convergence in $L_2$ and probability established

Sub-quadratic covariance growth ensures convergence

Clarifies folklore result for applied probability

Abstract

This brief pedagogical note re-proves a simple theorem on the convergence, in $L_2$ and in probability, of time averages of non-stationary time series to the mean of expectation values. The basic condition is that the sum of covariances grows sub-quadratically with the length of the time series. I make no claim to originality; the result is widely, but unevenly, spread bit of folklore among users of applied probability. The goal of this note is merely to even out that distribution.