On the almost sure minimal growth rate of partial sum maxima
type: Item
instance of: scholarly article
title: On the almost sure minimal growth rate of partial sum maxima
zbMATH Open document ID: 0857.60023
author: Michael J. Klass
author: Cun-Hui Zhang
published in: The Annals of Probability
publication date: 1996-02-01T00:00:00Z
review text: Let \(\{X_j\}_j\) be a sequence of i.i.d. random variables, \(S_n=\sum^n_{j=1} X_j\), and let \(\{a_n\}\) be an increasing sequence of positive constants tending to \(\infty\). This paper studies the almost lower limit of \(\max_{1\leq j \leq n}S_j/a_n\). It is shown that the lower limit is either 0 or \(\infty\) under mild conditions, and some integral tests to determine which is the case are given. Let \(\tau=\inf \{n\geq 1:S_n>0\}\) and \(\tau_-=\inf \{n\geq 1:S_n \leq 0\}\). Several inequalities are gotten that determine up to scale constants various quantities, which involve truncated moments of the ladder variables \(S_\tau\) and \(\tau\) under three different conditions: \(ES_\tau < \infty\), \(E|S_{\tau_-} |<\infty\) and \(X_j\) symmetric. Moments of ladder variables are also discussed. The problems discussed here have their origin in the desire on the part of many authors to refine, extend and achieve a deeper understanding of the strong law of large numbers and the law of the iterated logarithm. In 1965, Hirsch considered a specific question of concern here.
reviewed by: Chun Su
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 828369
zbMATH Keywords: almost sure limit
zbMATH Keywords: integral test
zbMATH Keywords: rate of escape
zbMATH Keywords: random walk
zbMATH Keywords: ladder variables
zbMATH Keywords: truncated moment
zbMATH Keywords: inequality
zbMATH Keywords: law of the iterated logarithm
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176988487
OpenAlex ID: W2080968832
DOI: 10.1214/AOP/1176988487
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description: scientific article; zbMATH DE number 828369