Uniform local probability approximations: Improvements on Berry-Esseen
type: Item
instance of: scholarly article
title: Uniform local probability approximations: Improvements on Berry-Esseen
zbMATH Open document ID: 0831.60048
author: Marjorie G. Hahn
author: Michael J. Klass
published in: The Annals of Probability
publication date: 1996-01-25T00:00:00Z
review text: Let \((X_i)\) be i.i.d. mean-zero, bounded r.v.s, \(S_n= X_1+\cdots+ X_n\). The authors prove under certain conditions results of the type \[ P(S_n\in \Gamma)= P(S_n/\sqrt{\text{var}(S_n)}\in \Gamma/\sqrt{\text{var}(S_n)})\approx |\Gamma|/ \sqrt{\text{var}(S_n)}, \] where \(|\cdot|\) denotes Lebesgue measure. This implies, in particular, that, whenever \(\Gamma\) intersects the interval between the \(\varepsilon\)th and the \((1- \varepsilon)\)th quantile of \(S_n\), \(P(S_n\in \Gamma)\) is proportional to the length of the interval over the standard deviation.
reviewed by: T. Mikosch
Mathematics Subject Classification ID: 60F99
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60E15
zbMATH DE Number: 796550
zbMATH Keywords: local limit theorem
zbMATH Keywords: Berry-Esseen inequality
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176988394
OpenAlex ID: W2059538756
DOI: 10.1214/AOP/1176988394
description: scientific article; zbMATH DE number 796550