Empirical processes with bounded \psi_1 diameter

TL;DR

This paper establishes sharp bounds on empirical processes indexed by squared functions using _1 diameters, extending results in asymptotic geometric analysis for isotropic, log-concave ensembles.

Contribution

It introduces bounds based on _1 diameters instead of _2, providing new insights into empirical processes and geometric analysis.

Findings

Bound on supremum depends on _1 diameter and b3_2 complexity

Optimal bounds on random diameters for function classes

Extension of geometric analysis results to log-concave ensembles

Abstract

We study the empirical process indexed by F^2=\{f^2 : f \in F\}, where F is a class of mean-zero functions on a probability space. We present a sharp bound on the supremum of that process which depends on the \psi_1 diameter of the class F (rather than on the \psi_2 one) and on the complexity parameter \gamma_2(F,\psi_2). In addition, we present optimal bounds on the random diameters \sup_{f \in F} \max_{|I|=m} (\sum_{i \in I} f^2(X_i))^{1/2} using the same parameters. As applications, we extend several well known results in Asymptotic Geometric Analysis to any isotropic, log-concave ensemble on R^n.