The central limit theorem for weighted empirical processes indexed by sets
type: Item
instance of: scholarly article
title: The central limit theorem for weighted empirical processes indexed by sets
zbMATH Open document ID: 0624.60051
DOI: 10.1016/0047-259X(87)90093-5
author: Kenneth S. Alexander
published in: Journal of Multivariate Analysis
publication date: 1987-01-01T00:00:00Z
review text: Let (S,\({\mathcal S})\) be a measurable space, let P be a probability measure on (S,\({\mathcal S})\) and let \(X_ i\) be the coordinate projections \((S^ N,{\mathcal S}^ N,P^ N)\to (S,{\mathcal S},P)\). Let \({\mathcal C}\subset {\mathcal S}\) be a Vapnik-Cervonenkis class [\textit{R. M. Dudley}, Ann. Probab. 6, 899-929 (1978; Zbl 0404.60016)]. Let \(\nu_ n=n^{- }\sum^{n}_{i=1}(\delta_{X_ i}-P)\). Let q be a non-negative function on [0,1], 0 at 0 and at 1, increasing near 0 and decreasing near 1. The classical Cibisov-O'Reilly theorem generalizes to weak convergence, uniform on \(C\in {\mathcal C}\), for \(\{\nu_ k(C)/q(P(C))\}\), but convergence is achieved only if the union of the sets in C with P(C)\(\leq t\) has probability tending to zero with t (besides other conditions) [the author, Probab. Theory Relat. Fields 75, 379-423 (1987; Zbl 0596.60029)]. For classes with ``too many small sets'' one may circumvent this difficulty by truncation, that is by asking: ``(1) for which q does there exist a sequence \(\{\gamma_ n\}\) of positive numbers such that the processes \(w_ n=\nu_ n(C)I(P(C))\geq \gamma_ n)/q(P(C))\) converge weakly to a Gaussian process uniformly in \(C\in {\mathcal C}\) and (2) given such a q, how slowly must \(\gamma_ n\to 0\) to ensure that weak convergence occurs?'' These questions are answered in the present article.
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 60G57
zbMATH DE Number: 4013656
zbMATH Keywords: central limit theorem
zbMATH Keywords: weighted empirical process
zbMATH Keywords: Vapnik-Cervonenkis class
zbMATH Keywords: weak convergence
reviewed by: Evarist Giné
Wikidata QID: Q101068605
MaRDI profile type: Publication
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full work available at URL: 0047-259x(87)90093-5
OpenAlex ID: W2066313192
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description: scientific article; zbMATH DE number 4013656