Sufficient and insufficient conditions for the stochastic convergence of Ces\`{a}ro means

TL;DR

This paper investigates the conditions under which the Cesàro mean of a sequence of random variables converges stochastically, highlighting that convergence rates for the sequence do not automatically imply similar rates for its Cesàro mean.

Contribution

The paper provides new sufficient conditions for the stochastic convergence of Cesàro means and clarifies when convergence rates transfer from sequences to their averages.

Findings

Rate of convergence in probability for a sequence does not imply the same for its Cesàro mean.

Identifies conditions under which Cesàro means converge stochastically.

Highlights common settings where these conditions are satisfied.

Abstract

We study the stochastic convergence of the Ces\`{a}ro mean of a sequence of random variables. These arise naturally in statistical problems that have a sequential component, where the sequence of random variables is typically derived from a sequence of estimators computed on data. We show that establishing a rate of convergence in probability for a sequence is not sufficient in general to establish a rate in probability for its Ces\`{a}ro mean. We also present several sets of conditions on the sequence of random variables that are sufficient to guarantee a rate of convergence for its Ces\`{a}ro mean. We identify common settings in which these sets of conditions hold.