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 $\Ex_{Ω_1} \Ex_{Ω_2}\max_{i,j}\abs{a_{ij}ξ_i(ω_1)η_j(ω_2)}$ 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 ($p\ge 2$) generates a truncated ${\ell }_p$-norm, especially $\Ex \max_{i}\abs{a_iξ_i}\sim \norm{(a_i)_i}_2$ for $p=2$. As far as we know, the generating distribution for ${\ell }_p$-norms with $p\ge 2$ has not been known up to now.

17 pages