# Sharper Sub-Weibull Concentrations

arXiv: https://arxiv.org/abs/2102.02450 · HTML: https://mlsanity.com/paper/2102.02450/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 35 references that are in the map.
- First posted on arXiv 4 Feb 2021.

## Builds on (most cited first, 8 of 8)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regression](https://mlsanity.com/paper/1804.02605/) (2018)
- [Bootstrapping Upper Confidence Bound](https://mlsanity.com/paper/1906.05247/) (2019)
- [Concentration Inequalities for Statistical Inference](https://mlsanity.com/paper/2011.02258/) (2020)
- [On the Non-asymptotic and Sharp Lower Tail Bounds of Random Variables](https://mlsanity.com/paper/1810.09006/) (2018)
- [Lecture Notes: Selected topics on robust statistical learning theory](https://mlsanity.com/paper/1908.10761/) (2019)
- [Elastic-net Regularized High-dimensional Negative Binomial Regression: Consistency and Weak Signals Detection](https://mlsanity.com/paper/1712.03412/) (2017)
- [Deterministic Inequalities for Smooth M-estimators](https://mlsanity.com/paper/1809.05172/) (2018)

## Built on by (most cited first, 8 of 8)

- [Multiplier Bootstrap-based Exploration](https://mlsanity.com/paper/2302.01543/) (2023)
- [On the best approximation by finite Gaussian mixtures](https://mlsanity.com/paper/2404.08913/) (2024)
- [Tight Non-asymptotic Inference via Sub-Gaussian Intrinsic Moment Norm](https://mlsanity.com/paper/2303.07287/) (2023)
- [Concentration inequalities for high-dimensional linear processes with dependent innovations](https://mlsanity.com/paper/2307.12395/) (2023)
- [A General Reduction for High-Probability Analysis with General Light-Tailed Distributions](https://mlsanity.com/paper/2403.02873/) (2024)
- [Asymptotic in a class of network models with an increasing sub-Gamma degree sequence](https://mlsanity.com/paper/2111.01301/) (2021)
- [Cost-Driven Representation Learning for Linear Quadratic Gaussian Control: Part II](https://mlsanity.com/paper/2603.07437/) (2026)
- [Tail-Aware Information-Theoretic Bounds for LLM Alignment under Heavy-Tailed Rewards](https://mlsanity.com/paper/2604.10727/) (2026)

## Abstract (arXiv)

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 ℓ2-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 Xii=1n without exponential-moment conditions.