Chevet-type inequalities for subexponential Weibull variables and estimates for norms of random matrices

Abstract

We prove two-sided Chevet-type inequalities for independent symmetric Weibull random variables with shape parameter $r\in [1,2]$. We apply them to provide two-sided estimates for operator norms from ${\ell }_p^n$ to ${\ell }_q^m$ of random matrices $(a_i{b}_j{X}_{i,j}{)}_{i\le m,j\le n}$, in the case when $X_{i,j}$'s are iid symmetric Weibull variables with shape parameter $r\in [1,2]$ or when $X$ is an isotropic log-concave unconditional random matrix. We also show how these Chevet-type inequalities imply two-sided bounds for maximal norms from ${\ell }_p^n$ to ${\ell }_q^m$ of submatrices of $X$ in both Weibull and log-concave settings.

17 pages. Corollary 3, Theorem 4, Corollary 5, Corollaries 10 & 12 and their proofs added, Conjecture 13 and Remark 14 added. Title and introduction changed