Uniform lower bounds for randomly stopped Banach space valued random sums
type: Item
instance of: scholarly article
title: Uniform lower bounds for randomly stopped Banach space valued random sums
zbMATH Open document ID: 0706.60046
author: Michael J. Klass
published in: The Annals of Probability
publication date: 1990-01-01T00:00:00Z
review text: Let \(X_ 1,X_ 2,..\). be independent, Banach space (B, \(\| \cdot \|)\) valued random elements with \(S_ n=X_ 1+X_ 2+...+X_ n\). Let T be any, possibly randomized, stopping time based on \(X_ n\) and let \(\phi\) (\(\cdot)\) be any nondecreasing continuous function on \([0,\infty)\) with \(\phi (0)=0\) such that \(\phi (cx)=c^{\alpha}\phi (x)\) for some fixed \(\alpha >0\) and for all \(x\geq 0\), \(c\geq 2.\) The author proves that there exists a universal constant \(c^*_{\alpha}\) depending only on \(\alpha >0\) such that \[ E\max_{1\leq n\leq \tilde T}\phi (\| S_ n\|)\leq c^*_{\alpha}E\max_{1\leq n\leq T}\phi (\| S_ n\|) \] where \(\tilde T\) has the same marginal distribution as T and is independent of \(\{X_ n\}\). An upper bound for \(c^*_{\alpha}\) is also constructed.
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60G40
Mathematics Subject Classification ID: 60E15
zbMATH DE Number: 4159783
zbMATH Keywords: maximum of normed partial sums
zbMATH Keywords: radom number of summands
zbMATH Keywords: stopping time
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176990858
OpenAlex ID: W2046033594
DOI: 10.1214/AOP/1176990858
description: scientific article; zbMATH DE number 4159783