Moment free deviation inequalities for linear combinations of independent random variables with power-type tails

Abstract

We present order of magnitude estimates for the quantiles of non-negative linear combinations of non-negative random variables, as well as deviation inequalities for general linear combinations of independent random variables, under the assumption that all random variables satisfy the same power-type tail bound on $\mathbb{P}\{\left|X_i\right|>t\}$ of the form $t^{-q}$, $t^{-q/2}$ or $t^{-q/2}(\ln t{)}^{q/2}$, for $q>2$. The third type is applicable in the nonlinear setting. In the situations we consider, these results improve on classical estimates of Nagaev.

29 pages. A significant amount of the material here was originally part of arXiv:2203.12523 and/or arXiv:1812.10938 and now stands as a paper on its own. This paper used to have the title 'Variations and extensions of the Gaussian concentration inequality, Part IIa: Linear case'