Empirical processes with bounded ψ₁ diameter

2010/05/05 by Shahar Mendelson, Mendelson, Shahar · 1 citation
Economics, Econometrics and Finance · Mathematics · #Stochastic processes and financial applications #math.FA #math.PR

paper · pdf · doi:10.48550/arxiv.1005.0816

arxiv created 2010/05/05 · arxiv updated 2010/05/06

Abstract

We study the empirical process indexed by F²=\f² : f ∈ 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 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 sup_{f ∈ F} max_|I|=m (∑_{i ∈ I} f²(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 Rⁿ.

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