Statistica Sinica 2(1992), 1-15

UNIFORM CONVERGENCE OF PROBABILITY

MEASURES ON CLASSES OF FUNCTIONS

P. J. Bickel and P. W. Millar

University of California, Berkeley

Abstract: Let p_n, p be probabilities, and F, F^* be collections of real functions. Simple conditions are derived under which the simple convergence of ∫ƒ(x)P_n(dx) to ∫ƒ(x)P(dx) for every ƒ in F^* implies uniform convergence over converges to 0. Several examples are discussed, some historical and some new.

Key words and phrases: Weak convergence of probabilities, uniform convergence of probabilities, Pólya class, Pólya's theorem, Glivenko-Cantelli theorem, dual Lipschi tz norm, bracketing, Vapnik-Cervonenkis class, convex sets, uniformity class, delta-tight.