Precise asymptotics - A general approach

ISSN： 0236-5294

Source： Acta Mathematica Hungarica, Vol.138, Iss.4, 2013-03, pp. : 365-385

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Abstract

The legendary 1947-paper by Hsu and Robbins, in which the authors introduced the concept of “complete convergence, generated a series of papers culminating in the like-wise famous Baum-Katz 1965-theorem, which provided necessary and sufficient conditions for the convergence of the series $sum_{n=1}^{infty}n^{r/p-2}P (|S_{n}| geqq varepsilon n^{1/p})$ for suitable values of r and p, in which S _n denotes the n-th partial sum of an i.i.d. sequence. Heyde followed up the topic in his 1975-paper where he investigated the rate at which such sums tend to infinity as ε↘0 (for the case r=2 and p=1). The remaining cases have been taken care later under the heading “precise asymptotics. An abundance of papers have since then appeared with various extensions and modifications of the i.i.d.-setting. The aim of the present paper is to show that the basis for the proof is essentially the same throughout, and to collect a number of examples. We close by mentioning that Klesov, in 1994, initiated work on rates in the sense that he determined the rate, as ε↘0, at which the discrepancy between such sums and their “Baum-Katz limit converges to a nontrivial quantity for Heyde's theorem. His result has recently been extended to the complete set of r- and p-values by the present authors.