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We study the stochastic convergence of the Ces\`{a}ro mean of a sequence ofrandom variables. These arise naturally in statistical problems that have asequential component, where the sequence of random variables is typicallyderived from a sequence of estimators computed on data. We show thatestablishing a rate of convergence in probability for a sequence is notsufficient in general to establish a rate in probability for its Ces\`{a}romean. We also present several sets of conditions on the sequence of randomvariables that are sufficient to guarantee a rate of convergence for itsCes\`{a}ro mean. We identify common settings in which these sets of conditionshold.

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