%0 Journal Article
%T Upper functions for $\mathbb{L}_{p}$-norms of Gaussian random fields
%+ Institut de Mathématiques de Marseille (I2M)
%A Lepski, Oleg
%< avec comité de lecture
%@ 1350-7265
%J Bernoulli
%I Bernoulli Society for Mathematical Statistics and Probability
%S bernoulli
%V 22
%N 2
%P 732-773
%8 2016
%D 2016
%R 10.3150/14-BEJ674
%K upper function
%K gaussian random field
%K metric entropy
%K Dudley's integral
%Z Mathematics [math]Journal articles
%X In this paper we are interested in finding upper functions for a collection of random variables { ξ ⃗ h p , ⃗ h ∈ H } , 1 ≤ p < ∞. Here ξ ⃗ h (x), x ∈ (−b, b) d , d ≥ 1 is a kernel-type gaussian random field and ∥ · ∥p stands for Lp-norm on (−b, b) d. The set H consists of d-variate vector-functions defined on (−b, b) d and taking values in some countable net in R d +. We seek a non-random family { Ψε (⃗ h) , ⃗ h ∈ H } such that E { sup ⃗ h∈H [ ξ ⃗ h p − Ψε (⃗ h)] + } q ≤ ε q , q ≥ 1, where ε > 0 is prescribed level.
%G English
%2 https://hal.science/hal-01265225/document
%2 https://hal.science/hal-01265225/file/gauss-L_p-norm_ineq-1new-rev_2.pdf
%L hal-01265225
%U https://hal.science/hal-01265225
%~ CNRS
%~ UNIV-AMU
%~ EC-MARSEILLE
%~ I2M
%~ I2M-2014-
%~ AMIDEX
%~ ANR