The law of the iterated logarithm for empirical processes on Vapnik- Červonenkis classes
type: Item
instance of: scholarly article
title: The law of the iterated logarithm for empirical processes on Vapnik- Červonenkis classes
zbMATH Open document ID: 0688.60021
DOI: 10.1016/0047-259X(89)90093-6
author: Kenneth S. Alexander
author: Michel Talagrand
published in: Journal of Multivariate Analysis
publication date: 1989-01-01T00:00:00Z
review text: The first author, Ann. Probab. 15, 178-203 (1987; Zbl 0624.60032), gave necessary and sufficient conditions for a VC-graph class \({\mathcal F}\) of functions to be P-Donsker (i.e. such that the empirical process based on P and indexed by \({\mathcal F}\) satisfies the CLT). Here the authors give necessary and sufficient conditions for the empirical process indexed by such classes to satisfy the law of iterated logarithm, both compact and bounded. The results are based on the recent remarkable characterization of the LIL for Banach space valued random variables of \textit{M. Ledoux} and \textit{M. Talagrand}, ibid. 16, 1242-1264 (1988; Zbl 0662.60008), which reduced the LIL to moment conditions and to convergence in probability to zero or stochastic boundedness of \(\| S_ n\| /\sqrt{n \log \log n}\). Because of this, entropy chaining proofs can be employed to obtain sharp results. The main part of the proof consists in showing that for VC-graph classes \({\mathcal F}\), if \(\tilde {\mathcal F}\) is the envelope of \({\mathcal F}\) then \[ E F^ 2(X) Log Log F(X)<\infty \quad implies\quad \| \nu_ n/\sqrt{Log Log n}\|_{{\mathcal F}}\to 0\quad in\quad probability, \] an entropy proof. (Here \(\nu_ n=n^{- 1/2}\sum^{n}_{i=1}(\delta_{X_ i}-P)\), \(X_ i\) i.i.d. (P), and \(Log(x)=\log (xve).)\) The main result is: Let \({\mathcal F}\) be a countably determined VC-graph class of functions on (T,\({\mathcal B},P)\) with envelope \(\tilde {\mathcal F}\) and let \({\mathcal F}_ P=\sup \{| f-Pf|:\quad f\in {\mathcal F}).\) Then the conditions \(\sup_{f\in F}Var(F(x))<\infty\) (\(\{\) f-Pf: \(f\in {\mathcal F}\}\) is totally bounded in \(L_ 2(P))\) and \(E F^ 2(X)/Log Log F(X)<\infty\) imply that the empirical process based on P and indexed by \({\mathcal F}\) satisfies the bounded law of iterated logarithm (the compact LIL), and conversely, if \({\mathcal F}\) satisfies the bounded (compact) LIL then these two conditions hold, but the second for \(F_ P\) instead of \(\tilde {\m…
reviewed by: Evarist Giné
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60B12
zbMATH DE Number: 4128156
zbMATH Keywords: Vapnik-Cervonenkis classes
zbMATH Keywords: empirical process
zbMATH Keywords: law of iterated logarithm
zbMATH Keywords: stochastic boundedness
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4128156