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arXiv · 2009.05974

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

Aurélien F. Bibaut·Alex Luedtke·Mark J. van der Laan
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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.

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Aurélien F. Bibaut, Alex Luedtke, Mark J. van der Laan. 2020-09-13. Sufficient and insufficient conditions for the stochastic convergence of Ces\`{a}ro means. https://arxiv.org/abs/2009.05974

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