On a problem of Csörgö and Révész
type: Item
instance of: scholarly article
title: On a problem of Csörgö and Révész
zbMATH Open document ID: 0684.60020
published in: The Annals of Probability
publication date: 1989-01-01T00:00:00Z
review text: Let \(\{X_ n\}\) be a sequence of i.i.d. random variables with mean 0 and variance 1. Let \(S_ n=X_ 1+...+X_ n\) and \(0<r<1\). It is well known that, by redefining \(\{S_ n\}\) on a richer probability space if necessary, there exists a standard Wiener process \(\{\) W(t), \(t>0\}\) such that \[ S_ n-W(n)=O((\log n)^{1/r})\quad a.s. \] when E exp(t\({}_ 0| X_ 1|^ r)<\infty\) for some \(t_ 0>0\). Answering a question posed in \textit{M. Csörgö} and \textit{P. Révész} [Strong approximations in probability and statistics. (1981)] the author proves that O((log n)\({}^{1/r})\) cannot be replaced by o((log n)\({}^{1/r})\). He also gives two theorems on the increments of \(S_ n\), supplementing some theorems in the book cited above.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60F17
zbMATH DE Number: 4121128
zbMATH Keywords: invariance principle
zbMATH Keywords: increments of partial sums
zbMATH Keywords: Strong approximations
author: Q. M. Shao
reviewed by: Toshio Mori
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176991428
OpenAlex ID: W1995465073
DOI: 10.1214/AOP/1176991428
description: scientific article; zbMATH DE number 4121128