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Universal Donsker classes and metric entropy. (English) Zbl 0631.60004

Author’s summary: Let (X,\({\mathcal A})\) be a measurable space and \({\mathcal F}^ a \)class of measurable functions on X. \({\mathcal F}\) is called a universal Donsker class if for every probability measure P on \({\mathcal A}\), the centered and normalized empirical measures \(n^{1/2}(P_ n-P)\) converge in law, with respect to uniform convergence over \({\mathcal F}\), to the limiting “Brownian bridge” process \(G_ p\). Then up to additive constants, \({\mathcal F}\) must be uniformly bounded.
Several nonequivalent conditions are shown to imply the universal Donsker property. Some are connected with the Vapnik-Červonenkis combinatorial condition on classes of sets, others with metric entropy. The implications between the various conditions are considered. Bounds are given for the metric entropy of convex hulls in Hilbert space.
Reviewer: V.Paulauskas

MSC:

60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F17 Functional limit theorems; invariance principles
60F05 Central limit and other weak theorems
60G17 Sample path properties
60G20 Generalized stochastic processes
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