Concentration inequalities, large and moderate deviations for self-normalized empirical processes
type: Item
instance of: scholarly article
title: Concentration inequalities, large and moderate deviations for self-normalized empirical processes
zbMATH Open document ID: 1021.60013
author: Bernard Bercu
author: Elisabeth Gassiat
author: Emmanuel Rio
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: Let \((X_n)_{n\geq 1}\) be a sequence of independent and identically distributed random variables in some measurable space \(X\), and let \(\mathcal{F}\) be a class of real-valued measurable functions on \(X\) which are centered and normalized, i.e. \(E(f(X_1))=0\) and \(E(f^2(X_1))=1\) for all \(f\in\mathcal{F}\). Let \(P_n=n^{-1}\sum_{k=1}^n\delta_{X_k}\) denote the empirical measure based on the observations \(X_1,\dots,X_n\), and consider the self-normalized empirical process \(W_n(f)=P_n(f)/\sqrt{P_n(f^2)}\) indexed by \(f\in\mathcal{F}\). For the supremum \({\mathcal W}_n=\sup_{f\in\mathcal{F}}W_n(f)\) of \(W_n\) an exponential concentration inequality is proven for certain classes \(\mathcal{F}\) of unbounded functions. This inequality is applied to establish a moderate deviation principle for \({\mathcal W}_n\). Moreover, results on large deviations of \({\mathcal W}_n\) are obtained for some parametric classes \(\mathcal{F}\) of unbounded functions.
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 62G30
zbMATH DE Number: 1906095
zbMATH Keywords: maximal inequalities
zbMATH Keywords: self-normalized sums
zbMATH Keywords: empirical processes
zbMATH Keywords: concentration inequalities
zbMATH Keywords: large deviations
zbMATH Keywords: moderate deviations
zbMATH Keywords: logarithmic Sobolev inequalities
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1039548367
OpenAlex ID: W2032366027
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DOI: 10.1214/AOP/1039548367
description: scientific article; zbMATH DE number 1906095