On the Maximum of Random Variables on Product Spaces

TL;DR

This paper derives estimates for the expected maximum of products of independent p-stable and q-stable random variables on product spaces, linking them to specific mixed norm structures, and introduces a new distribution-based method for generating lp norms.

Contribution

It provides new bounds for maxima of products of stable variables in terms of mixed norms and introduces a novel distribution for generating lp norms with p or p  2.

Findings

Estimates for maxima of products of p-stable and q-stable variables in terms of mixed norms.

Identification of a distribution that generates lp norms for p or p  2.

Connection between lp norms and maxima of log-mp gamma distributed variables.

Abstract

Let $\xi_i$, $i=1,...,n$, and $\eta_j$, $j=1,...,m$ be iid p-stable respectively q-stable random variables, $1<p<q<2$. We prove estimates for $\Ex_{\Omega_1} \Ex_{\Omega_2}\max_{i,j}\abs{a_{ij}\xi_i(\omega_1)\eta_j(\omega_2)}$ in terms of the $\ell_p^m(\ell_q^n)$-norm of $(a_{ij})_{i,j}$. Additionally, for p-stable and standard gaussian random variables we prove estimates in terms of the $\ell_p^m(\ell_{M_{\xi}}^n)$-norm, $M_{\xi}$ depending on the Gaussians. Furthermore, we show that a sequence $\xi_i$, $i=1,...,n$ of iid $\log-\gamma(1,p)$ distributed random variables ($p\geq 2$) generates a truncated $\ell_p$-norm, especially $\Ex \max_{i}\abs{a_i\xi_i}\sim \norm{(a_i)_i}_2$ for $p=2$. As far as we know, the generating distribution for $\ell_p$-norms with $p\geq 2$ has not been known up to now.