Empirical processes with a bounded ψ 1 diameter Shahar

We study the empirical process supf∈F |N−1 ∑N i=1 f (Xi) − Ef2|, where F is a class of mean-zero functions on a probability space (Ω, μ) and (Xi)i=1 are selected independently according to μ. We present a sharp bound on this supremum that depends on the ψ1 diameter of the class F (rather than on the ψ2 one) and on the complexity parameter γ2(F, ψ2). In addition, we present optimal bounds on the random diameters supf∈F max|I|=m( ∑ i∈I f (Xi)) using the same parameters. As applications, we extend several well known results in Asymptotic Geometric Analysis to any isotropic, log-concave ensemble on R.