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 $\{\Vert {\xi }_{\vec{h}}{\Vert }_p,\vec{h}\in \mathrm{H}\},1\le p<\infty$. Here ${\xi }_{\vec{h}}(x),x\in (-b,b{)}^d,d\ge 1$ is a kernel-type gaussian random field and $\Vert \cdot {\Vert }_p$ 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 $\left\{{\Psi }_{\alpha }\left(\vec{h}\right),\vec{h}\in \mathrm{H}\right\}$ such that $E\left\{{\sup }_{\vec{h}\in \mathrm{H}}\right[\Vert {\xi }_{\vec{h}}{\Vert }_p-{\Psi }_{\alpha }\left(\vec{h}\right){]}_{+}{\}}^q\le {\alpha }^q,q\ge 1,$ where $\alpha >0$ is prescribed level.

References in corpus (4)

• Concentration inequalities and asymptotic results for ratio type empirical processes
• Risk hull method and regularization by projections of ill-posed inverse problems
• Structural adaptation via $L_p$-norm oracle inequalities
• Upper functions for positive random functionals