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Local invariance principles and their application to density estimation. (English) Zbl 0794.60019
Let \(x_ 1,\dots,x_ n\) be independent random variables with uniform distribution over \([0,1]^ d\), and \(X^{(n)}\) be the centered and normalized empirical process associated to \(x_ 1, \dots,x_ n\). Given a Vapnik-Chervonenkis class \({\mathcal S}\) of bounded functions from \([0,1]^ d\) into \(\mathbb{R}\) of bounded variation, we apply the one- dimensional dyadic scheme of Komlós, Major and Tusnády to get the best possible rate in Dudley’s uniform central limit theorem for the empirical process \(\{X^{(n)} (h):h \in {\mathcal S}\}\). When \({\mathcal S}\) fulfills some extra condition, we prove there exists some sequence \(B_ n\) of Brownian bridges indexed by \({\mathcal S}\) such that \[ \sup_{h \in {\mathcal S}} | X^{(n)} (h)-B_ n(h) |=O \bigl( n^{-1/2} \log n \vee n^{-1/(2d)} \sqrt {K({\mathcal S}) \log n} \bigr) \text{ a.s.}, \] where \(K({\mathcal S})\) denotes the maximal variation of the elements of \({\mathcal S}\). This result is then applied to maximal deviations distributions for kernel density estimators under minimal assumptions on the sequence of bandwith parameters. We also derive some results concerning strong approximations for empirical processes indexed by classes of sets with uniformly small perimeter. For example, it follows from J. Beck’s paper [Z. Wahrscheinlichkeitstheorie Verw. Geb. 70, 289-306 (1985; Zbl 0554.60037)] that the above result is optimal, up to a possible factor \(\sqrt {\log n}\), when \({\mathcal S}\) is the class of Euclidean balls with radius less than \(r\).
Reviewer: E.Rio (Orsay)

MSC:
60F17 Functional limit theorems; invariance principles
62G05 Nonparametric estimation
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