Empirical Processes with a Bounded Ψ 1 Diameter

Abstract

We study the empirical process $${{\rm sup}_{f \in F}|N^{-1}\sum_{i=1}^{N}\,f^{2}(X_i)-\mathbb{E}f^{2}|}$$ , where F is a class of mean-zero functions on a probability space (Ω, μ), and $${(X_{i})_{i =1}^N}$$ are selected independently according to μ. We present a sharp bound on this supremum that depends on the Ψ 1 diameter of the class F (rather than on the Ψ 2 one) and on the complexity parameter γ 2(F,Ψ 2). In addition, we present optimal bounds on the random diameters $${{\rm sup}_{f \in F} {\rm max}_{|I|=m}\left(\sum_{i \in I} f^{2}(X_{i})\right)^{1/2}}$$ using the same parameters. As applications, we extend several well-known results in Asymptotic Geometric Analysis to any isotropic, log-concave ensemble on $${\mathbb{R}^n}$$ .