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On the Maximum of Random Variables on Product Spaces

Let $ξ_i$, $i=1,...,n$, and $η_j$, $j=1,...,m$ 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 $\ell_p^m(\ell_{M_ξ}^n)$-norm, $M_ξ$ depending on the Gaussians. Furthermore, we show that a sequence $ξ_i$, $i=1,...,n$ of iid $\log-γ(1,p)$ distributed random variables ($p\geq 2$) 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 $p\geq 2$ has not been known up to now.