Sharper Sub-Weibull Concentrations

TL;DR

This paper develops sharper concentration inequalities for sums of independent sub-Weibull variables, improving bounds and introducing new tools for statistical estimation and random matrix theory under heavy-tailed conditions.

Contribution

It introduces sharper, constants-specified concentration bounds for sub-Weibull sums, a new sub-Weibull parameter, and applies these results to regression, random matrices, and robust estimation.

Findings

Sharper concentration inequalities with improved constants.

New sub-Weibull parameter enabling tight bounds.

Application to negative binomial regression and random matrix theory.

Abstract

Constant-specified and exponential concentration inequalities play an essential role in the finite-sample theory of machine learning and high-dimensional statistics area. We obtain sharper and constants-specified concentration inequalities for the sum of independent sub-Weibull random variables, which leads to a mixture of two tails: sub-Gaussian for small deviations and sub-Weibull for large deviations from the mean. These bounds are new and improve existing bounds with sharper constants. In addition, a new sub-Weibull parameter if the italic should be retained. Please check the whole text. is also proposed, which enables recovering the tight concentration inequality for a random variable (vector). For statistical applications, we give an $\ell_2$-error of estimated coefficients in negative binomial regressions when the heavy-tailed covariates are sub-Weibull distributed with sparse structures, which is a new result for negative binomial regressions. In applying random matrices, we derive non-asymptotic versions of Bai-Yin's theorem for sub-Weibull entries with exponential tail bounds. Finally, by demonstrating a sub-Weibull confidence region for a log-truncated Z-estimator without the second-moment condition, we discuss and define the sub-Weibull type robust estimator for independent observations $\{X_i\}_{i=1}^{n}$ without exponential-moment conditions.