A limit theorem for the moment of self-normalized sums
type: Item
instance of: scholarly article
title: A limit theorem for the moment of self-normalized sums
zbMATH Open document ID: 1186.60029
DOI: 10.1155/2009/957056
published in: Journal of Inequalities and Applications
publication date: 2009-11-03T00:00:00Z
full work available at URL: https://eudml.org/doc/117849
review text: Consider a zero mean random walk \(\{S_n\}\). Hsu-Robbins-Spitzer-Baum-Katz type theorems concern finiteness of Borel-Cantelli sums related to deviations from the mean, that is, of \(P(|S_n|>\varepsilon g(n))\), where \(g(n)\) typically is a (suitable) power of \(n\) possibly with some logartihm or iterated logarithm added, depending on whether one considers LLN, LIL, and so on. Theorems called ``Precise estimates\dots'' concern, properly scaled, limits of such sums as \(\varepsilon\searrow\) (to \(0\) or \(\sigma\sqrt{2}\), etc., depending on the kind of problem at hand). Such results can be extended to moments, in which case probabilities are replaced by centered expectations, and, also, as in the present paper, to self-normalized sums. The usual objects of investigation in the context of precise asymptotics are asymptotics concerning probabilities. In this paper the authors establish corresponding results for expected values of the counting process associated with the sequence of record times, extending earlier ones for probabilities.
reviewed by: Allan Gut
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60E07
zbMATH DE Number: 5624608
zbMATH Keywords: moments
zbMATH Keywords: self-normalized sums
zbMATH Keywords: probabilities
zbMATH Keywords: expected values
author: Qing Pei Zang
MaRDI profile type: Publication
OpenAlex ID: W2121904795
Wikidata QID: Q59248383
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description: scientific article; zbMATH DE number 5624608