On the Maximum of Random Variables on Product Spaces

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.