Precise asymptotics: A general approach
2013 (English)In: Acta Mathematica Hungarica, ISSN 0236-5294, E-ISSN 1588-2632, Vol. 138, no 4, 365-385 p.Article in journal (Refereed) Published
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 Sigma(infinity)(n-1) n(r/p-2)P(vertical bar S-n vertical bar >= epsilon 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 epsilon SE arrow 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 epsilon SE arrow 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.
Place, publisher, year, edition, pages
2013. Vol. 138, no 4, 365-385 p.
law of large numbers, Hsu-Robbins, Baum-Katz, convergence rate
IdentifiersURN: urn:nbn:se:uu:diva-196514DOI: 10.1007/s10474-012-0236-1ISI: 000314691900005OAI: oai:DiVA.org:uu-196514DiVA: diva2:610942