Upper functions for positive random functionals. I. General setting and Gaussian random functions |
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Authors: | O Lepski |
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Institution: | 14100. Laboratoire d’Analyse, Topologie, Probab., Univ. Aix-Marseille, Marseille, France
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Abstract: | 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 → ?+ 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. |
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