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

TL;DR

This paper develops two-sided Chevet-type inequalities for independent symmetric Weibull variables with shape $r\in[1,2]$, bridging Gaussian and exponential tails. It uses these bounds to obtain sharp two-sided estimates for operator norms $\|X\|_{\ell_p^n\to\ell_q^m}$ of tensor-structured random matrices $X_{ij}=a_i b_j X_{ij}$ and extends the analysis to isotropic unconditional log-concave matrices. The authors derive explicit bounds for submatrices, showing how maximal $\ell_p\to\ell_q$ norms over all $k\times l$ submatrices behave, and discuss implications for conjectures in the weighted Weibull and log-concave settings. The results unify and extend known Gaussian and exponential bounds to the Weibull and log-concave frameworks, with potential applications in high-dimensional probability, random matrix theory, and convex geometry. Overall, the work provides a robust toolkit for estimating norms and tails of structured random matrices beyond the Gaussian paradigm.

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 $(a_ib_jX_{i,j})_{i\le m, j\le n}$, 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.

Theorems & Definitions (36)

• Theorem 1
• Corollary 2
• Corollary 3
• Theorem 4
• Corollary 5
• Corollary 6
• proof : Proof of Theorem \ref{['thm:chevetWeibull']}
• Lemma 7
• proof
• Lemma 8