Upper functions for $L_p$-norm of gaussian random fields

Abstract

In this paper we are interested in finding upper functions for a collection of random variables $\big\{\big\|\xi_{\vec{h}}\big\|_p, \vec{h}\in\mathrm{H}\big\}, 1\leq p<\infty$. Here $\xi_{\vec{h}}(x), x\in(-b,b)^d, d\geq 1$ is a kernel-type gaussian random field and $\|\cdot\|_p$ stands for $L_p$-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 $R^d_+$. We seek a non-random family $\left\{\Psi_\alpha\big(\vec{h}\big),\;\;\vec{h}\in\mathrm{H}\right\}$ such that $ E\big\{\sup_{\vec{h}\in\mathrm{H}}\big[\big\|\xi_{\vec{h}}\big\|_p-\Psi_\alpha\big(\vec{h}\big)\big]_+\big\}^q\leq \alpha^q,\; q\geq 1, $ where $\alpha>0$ is prescribed level.