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 → 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.
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Lepski, O. Upper functions for positive random functionals. I. General setting and Gaussian random functions. Math. Meth. Stat. 22, 1–27 (2013). https://doi.org/10.3103/S1066530713010018
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DOI: https://doi.org/10.3103/S1066530713010018