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.