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

2022/03/17 by Shalizi, Cosma Rohilla
#Data Analysis #FOS: Mathematics #FOS: Physical sciences #Probability (math.PR) #Statistics and Probability (physics.data-an)

paper · doi:10.48550/arxiv.2203.09085

Abstract

This brief pedagogical note re-proves a simple theorem on the convergence, in L₂ 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.

Related

• Convergence of multiple ergodic averages
• The Failure of the Ergodic Assumption
• Absolute and Unconditional Convergence of Series of Ergodic Averages and Lebesgue Derivatives
• Pointwise ergodic theorems with rate and application to limit theorems for stationary processes
• Fluctuation of ergodic averages and other stochastic processes