Empirical processes with bounded \psi_1 diameter
Functional Analysis
2010-05-06 v1 Probability
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.
Cite
@article{arxiv.1005.0816,
title = {Empirical processes with bounded \psi_1 diameter},
author = {Shahar Mendelson},
journal= {arXiv preprint arXiv:1005.0816},
year = {2010}
}