Abstract

Giné and Zinn have given a Gaussian characterization of classes of functions for which the empirical process satisfies the central limit theorem uniformly over all distributions of the underlying variables. A central object that arises in this characterization is the uniform Gaussian width, which can be upper bounded by the Koltchinskii-Pollard uniform entropy integral. In simple examples, however, the uniform Gaussian width proves to behave in a strictly better manner than might be expected from such computations. This phenomenon is not due to the inefficiency of classical chaining arguments, as might be expected in view of the majorizing measure theory, but rather due to the fact that the uniform entropy can grow at a strictly faster rate than the entropy with respect to any fixed distribution. In this talk I will aim to explain this phenomenon, its connection with combinatorial parameters, and whatever understanding I have at the present time (which is very limited) about the occurence of such behavior in general settings.