Open Access
May 2016 Upper functions for $\mathbb{L}_{p}$-norms of Gaussian random fields
Oleg Lepski
Bernoulli 22(2): 732-773 (May 2016). DOI: 10.3150/14-BEJ674


In this paper, we are interested in finding upper functions for a collection of random variables $\{\|\xi_{\vec{h}}\|_{p},\vec{h}\in\mathrm{H}\},1\leq p<\infty$. Here $\xi_{\vec{h}}(x),x\in(-b,b)^{d},d\geq1$ is a kernel-type Gaussian random field and $\|\cdot\|_{p}$ stands for $\mathbb{L}_{p}$-norm on $(-b,b)^{d}$. The set $\mathrm{H}$ consists of $d$-variate vector-functions defined on $(-b,b)^{d}$ and taking values in some countable net in $\mathbb{R}^{d}_{+}$. We seek a non-random family $\{\Psi_{\varepsilon}(\vec{h}),\vec{h}\in\mathrm{H}\}$ such that $\mathbb{E}\{\sup_{\vec{h}\in\mathrm{H}}[\|\xi_{\vec{h}}\|_{p}-\Psi_{\varepsilon}(\vec{h})]_{+}\}^{q}\leq\varepsilon^{q},q\geq1$, where $\varepsilon>0$ is prescribed level.


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Oleg Lepski. "Upper functions for $\mathbb{L}_{p}$-norms of Gaussian random fields." Bernoulli 22 (2) 732 - 773, May 2016.


Received: 1 December 2013; Revised: 1 July 2014; Published: May 2016
First available in Project Euclid: 9 November 2015

zbMATH: 1335.60077
MathSciNet: MR3449799
Digital Object Identifier: 10.3150/14-BEJ674

Keywords: Dudley’s integral , Gaussian random field , Metric entropy , upper function

Rights: Copyright © 2016 Bernoulli Society for Mathematical Statistics and Probability

Vol.22 • No. 2 • May 2016
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