Tight Concentration Inequality for Sub-Weibull Random Variables with Generalized Bernstien Orlicz norm

TL;DR

This paper develops tight concentration inequalities for sub-Weibull random variables using Generalized Bernstein-Orlicz norms, extending classical bounds to heavy-tailed distributions and improving sample complexity in graphical model inference.

Contribution

It introduces generalized Bernstein's inequalities and Rosenthal-type bounds for sub-Weibull variables, with proofs of tightness and applications to high-dimensional inference.

Findings

Derived tight concentration bounds for sub-Weibull variables.

Established lower bounds confirming the bounds' optimality.

Improved sample complexity bounds in graphical model inference.

Abstract

Recent development in high-dimensional statistical inference has necessitated concentration inequalities for a broader range of random variables. We focus on sub-Weibull random variables, which extend sub-Gaussian or sub-exponential random variables to allow heavy-tailed distributions. This paper presents concentration inequalities for independent sub-Weibull random variables with finite Generalized Bernstein-Orlicz norms, providing generalized Bernstein's inequalities and Rosenthal-type moment bounds. The tightness of the proposed bounds is shown through lower bounds of the concentration inequalities obtained via the Paley-Zygmund inequality. The results are applied to a graphical model inference problem, improving previous sample complexity bounds.