# Existence of Moments of Empirical Versions of Hsu—Robbins—Baum—Katz Series

## DOI:

https://doi.org/10.20535/1810-0546.2016.4.72344## Keywords:

Complete convergence for sums of independent identically distributed random variables, Empirical Hsu-Robbins and Baum-Katz series, Multi-indexed sums, Regularly varying weights## Abstract

**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.

## References

P.L. Hsu and H. Robbins, “Complete convergence and the law of large numbers”, *Proc. Nat. Acad. Sci. USA*, vol. 33, pp. 25–31, 1947.

L.E. Baum and M. Katz, “Convergence rates in the law of large numbers”, *Trans. Amer. Math. Soc.*, vol. 120, pp. 108–123, 1965.

J.A. Davis, “Convergence rates for probabilities of moderate deviations”, *Ann. Math. Staist.*, vol. 39, pp. 2016–2028, 1968.

C.C. Heyde, and V.K. Rohatgi, “A pair of complementary theorems on convergence rates in the law of large numbers”, *Proc. Camb. Phil. Soc.*, vol. 63, pp. 73–82, 1967.

F. Spitzer, “A combinatorial lemma and its applications to probability theory”, *Trans. Amer. Math. Soc*., vol. 82, no. 2, pp. 323–339, 1956.

A. Gut, “Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices”, *Ann. Probab.*, vol. 6, pp. 469–482, 1978.

A. Gut, “Convergence rates for probabilities of moderate deviations for sums of random variables with multidimensional indices”, *Ann. Probab.*, vol. 8, pp. 298–313, 1980.

A. Gut, “Strong laws for independent identically distributed random variables indexed by a sector”, *Ann. Prob.*, vol. 11, pp. 569–577, 1983.

A. Gut and U. Stadtmüller, “An asymmetric Marcinkiewicz–Zygmund LLN for random fields”, *Stat. Probab. Lett.*, vol. 79, pp. 1016–1020, 2009.

O.I. Klesov, “The strong law of large numbers for multiple sums of independent identically distributed random variables”, *Math. Notes*, vol. 38, pp. 1006–1014, 1986.

R.T. Smythe, “Strong laws of large numbers for *r*-dimensional arrays of random variables”, *Ann. Probab.*, vol. 1, pp. 164–170, 1973.

R.T. Smythe, “The sums of independent random variables on the partially ordered sets”, *Ann. Probab.*, vol. 2, pp. 906–917, 1974.

O.I. Klesov, *Limit Theorems for Multi-Indexed Sums of Random Variables*.New York: Springer, 2014.

J. Slivka and N.C. Severo, “On the strong law of large numbers”, *Proc. Nat. Acad. Sci. U.S.A.*, vol. 24, pp. 729–734, 1970.

J. Marcinkiewicz and A. Zygmund, “Sur les fonctions indépendantes”, *Fund. Math.*, vol. 29, pp. 60–90, 1937.

P. Erdős, “On a theorem of Hsu and Robbins”, *Ann. Math. Statist.*, vol. 20, pp. 286–291; vol. 21, pp. 138, 1949.

A. Gut, *Probability: A Graduate Course*, 2nd ed.New York: Springer-Verlag, 2013.

G.H. Hardy and E.M. Wright, *An Introduction to the Theory of Numbers*. 3rd ed. Oxford, UK: OxfordUniversity Press, 1954.

O. Klesov and U. Stadtmüller, “Existence of moments in the Hsu-Robbins-Erdös theorem”, *Annales Uni. Sci. Budapest, Sect. Comp.*, vol. 39, pp. 271–278, 2013.

N.H. Bingham *et al*., *Regular Variation*. Cambridge, UK: Cambridge University Press, 1987.

V.V. Buldygin *et al*., *Pseudo-Regular Variation in Generalized Renewal Theory*. Kyiv, Ukraine: TBiMC, 2012.

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