Concentration Inequalities for Sub-Weibull Random Tensors

Abstract

We extend the theory of concentration inequalities to simple random tensors with heavy-tailed coefficients. Specifically, we consider the class of sub-Weibull distributions ${\mathcal{S}}_{\alpha }$ for $\alpha \in [1,2]$. We establish concentration bounds for Euclidean functions of such tensors, exhibiting a phase transition between sub-gaussian and heavy-tailed regimes. Our results rely on a new Generalized Maximal Inequality for products of heavy-tailed random variables and a martingale analysis using Nagaev-type inequalities.