Strong moderate deviation theorems
type: Item
instance of: scholarly article
title: Strong moderate deviation theorems
zbMATH Open document ID: 0757.60015
author: Wilbert C. M. Kallenberg
author: Teresa Ledwina
author: T. Inglot
published in: The Annals of Probability
publication date: 1993-01-16T00:00:00Z
review text: Let \(T_ n\) be a sequence of random variables such that there are r.v. \(W_ n\) satisfying the following conditions \[ P(\sqrt n| T_ n-W_ n|>c_ 1\lg n+x)\leq c_ 2 e^{-c_ 3x} \] for all large enough \(n\) and all \(0<x<c_ 4 n^{1/3}\) and assume that \[ P(W_ n>x)=\exp(-2^{- 1} ax^ 2+g(x)) \] with \(g(x)\) satisfying certain regularity assumptions. The authors show that then \(\lim_{n\to\infty}(P(T_ n>x_ n)/P(W_ n>x_ n))=1\), if \(x_ n\to\infty\) and \(x_ n=o(n^{-1/6})\). The result is applied to weighted Kolmogorov-Smirnov statistics, generalized Cramér-von Mises statistics, chi-square statistics, Watson statistics, quadratic statistics and \(L\)- statistics.
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60G30
zbMATH DE Number: 89798
zbMATH Keywords: moderate deviations
zbMATH Keywords: Kolmogorov-Smirnov statistics
zbMATH Keywords: Cramér-von Mises statistics
zbMATH Keywords: chi-square statistics
zbMATH Keywords: Watson statistics
zbMATH Keywords: quadratic statistics
zbMATH Keywords: \(L\)-statistics
reviewed by: Włodek Bryc
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176989814
OpenAlex ID: W2039987920
DOI: 10.1214/AOP/1176989814
description: scientific article; zbMATH DE number 89798