Classes of functions related to VC properties

Abstract

The notion of Vapnik-Chervonenkis (VC) class of sets can be extended to classes of functions in a few ways. Under further hypotheses, central limit theorems for empirical measures can be proved uniformly over such classes. Specific such classes on Euclidean spaces can be used to show the existence of location vector and scatter matrix functionals, replacing mean vectors and covariance matrices, but on classes of probability measures P that are weakly dense, weakly open, and so contain arbitrarily heavy-tailed distributions.