Upper functions for positive random functionals. I. General setting and Gaussian random functions

In this paper we are interested in finding upper functions for a collection of real-valued random variables {Ψ( χ θ ), θ ∈ Θ}. Here { χ θ , θ ∈ Θ} is a family of continuous random mappings, Ψ is a given sub-additive positive functional and Θ is a totally bounded subset of a metric space. We seek a nonrandom function U : Θ → ℝ + such that sup θ ∈Θ {Ψ( χ θ ) − U ( θ )} + is “small” with prescribed probability. We apply the results obtained in the general setting to the variety of problems related to Gaussian random functions and empirical processes.