Entropy conditions for \(L_{r}\)-convergence of empirical processes
type: Item
instance of: scholarly article
title: Entropy conditions for \(L_{r}\)-convergence of empirical processes
zbMATH Open document ID: 1170.60004
author: Andrea Caponnetto
author: Ernesto De Vito
author: Massimiliano Pontil
published in: Advances in Computational Mathematics
publication date: 2009-05-04T00:00:00Z
review text: Let \(P_n\) be the empirical measure for a sample of \(n\) independent random elements with joint distribution \(P\). Furthermore, let \({\mathcal F}\) be a class of real-valued (bounded) functions defined on the sample space. Then it is known that \(\sup_{f\in {\mathcal F}}|P_nf - Pf| \to 0\) almost surely provided that \({\mathcal F}\) admits a finite entropy integral. In the present paper the appropriate entropy is discussed to ensure convergence of \(\int_{{\mathcal F}}|P_nf-Pf|^r d\mu(f)\) to zero, where \(\mu\) is a given measure on \({\mathcal F}\).
reviewed by: Winfried Stute
Mathematics Subject Classification ID: 60B10
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60G51
zbMATH DE Number: 5550554
zbMATH Keywords: Empirical processes
zbMATH Keywords: Uniform entropy
zbMATH Keywords: Rademacher averages
zbMATH Keywords: Glivenko-Cantelli classes
Wikidata QID: Q60700499
MaRDI profile type: Publication
full work available at URL: s10444-008-9072-9
OpenAlex ID: W1993177744
cites work: Scale-sensitive dimensions, uniform convergence, and learnability
cites work: Q4881152
cites work: Q3217346
cites work: Uniform Central Limit Theorems
cites work: Q4741476
cites work: Sous-espaces d'espaces vectoriels topologiques et noyaux associés. (Noyaux reproduisants.)
cites work: Weak convergence and empirical processes. With applications to statistics
cites work: Q4261789
cites work: On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities
cites work: Q4770512
cites work: Necessary and Sufficient Conditions for the Uniform Convergence of Means to their Expectations
DBLP publication ID: journals/adcm/CaponnettoVP09
DOI: 10.1007/S10444-008-9072-9
description: scientific article; zbMATH DE number 5550554