Sufficient and insufficient conditions for the stochastic convergence of Cesàro means

09/13/2020

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by   Aurélien F. Bibaut, et al.

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We study the stochastic convergence of the Cesàro 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 Cesàro 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 Cesàro mean. We identify common settings in which these sets of conditions hold.