Precise asymptotics: a general approach
2011 (English)Report (Other academic)
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 su_cient conditions for the convergence of the series P1 n=1 nr=p2P(jSnj _ "n1=p) for suitable values of r and p, in which Sn denotes the n-th partial sum of an i.i.d. sequence. Heyde followed up the topic in his 1975-paper in that he investigated the rate at which such sums tend to in_nity as " & 0 (for the case r = 2 and p = 1). The remaining cases have been taken care of later under the heading \precise asymptotics". An abundance of papers have since then appeared with various extensions and modi_cations 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 " & 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
2011. , 15 p.
U.U.D.M. report / Uppsala University, Department of Mathematics, ISSN 1101-3591 ; 2011:19
Probability Theory and Statistics
Research subject Mathematical Statistics
IdentifiersURN: urn:nbn:se:uu:diva-160647OAI: oai:DiVA.org:uu-160647DiVA: diva2:452169