Existence of Moments of Empirical Versions of Hsu?Robbins?Baum?Katz Series

Background. We study the so called empirical versions of Hsu?Robbins and Baum?Katz series that are the basic notion of the classical theory of complete convergence. Objective. The aim of the paper is to find necessary and sufficient conditions for the almost sure convergence of empirical Baum?Katz series. These conditions are expressed in terms of the existence of certain moments of the underlying random variables. Methods. For proving our results we develop some new technique based on truncation and studying the truncated random variables. A sufficient ingredient of our approach is to show that the behavior of the truncated versions and the original ones is the same. Despite some similarity between the original series and its empirical version, the methods for achieving the results are quite different. Results. We find necessary and sufficient conditions for the existence of higher moments of empirical versions. A special attention is paid to the case of multi-indexed sums. The latter case differs essentially from the one-dimensional case, since the space of indices is not completely ordered and thus any approach based on the first hitting moment does not work here. Conclusions. The results obtained in the paper may serve as a base for further studies of empirical versions that could be used in statistical procedures of estimating an unknown variance.