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

TL;DR

This paper derives deviation inequalities and quantile estimates for linear combinations of independent random variables with power-type tails, extending classical results and applicable in nonlinear contexts.

Contribution

It introduces new deviation inequalities and quantile estimates for sums of independent variables with power-type tails, including nonlinear cases, improving upon classical bounds.

Findings

Provides order of magnitude estimates for quantiles.

Establishes deviation inequalities under power-type tail bounds.

Extends results to nonlinear settings.

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 $\mathbb{P}\{\left\vert X_i\right\vert>t\}$ 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.