Upper functions for $\mathbb{L}_{p}$-norms of Gaussian random fields

Abstract

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.