The central limit theorem for weighted empirical processes indexed by sets
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.
scientific article; zbMATH DE number 4013656
scientific article