Sharp Concentration Results for Heavy-Tailed Distributions

TL;DR

This paper develops sharp concentration and large deviation inequalities for sums of heavy-tailed i.i.d. random variables, extending existing results and simplifying analysis through truncation methods.

Contribution

It introduces a unified framework for concentration inequalities applicable to heavy-tailed distributions, including new results for heavier tails.

Findings

Recovered concentration results for subWeibull variables

Derived new large deviation bounds for heavier tails

Simplified proofs using truncation techniques

Abstract

We obtain concentration and large deviation for the sums of independent and identically distributed random variables with heavy-tailed distributions. Our concentration results are concerned with random variables whose distributions satisfy $\mathbb{P}(X>t) \leq {\rm e}^{- I(t)}$, where $I: \mathbb{R} \rightarrow \mathbb{R}$ is an increasing function and $I(t)/t \rightarrow \alpha \in [0, \infty)$ as $t \rightarrow \infty$. Our main theorem can not only recover some of the existing results, such as the concentration of the sum of subWeibull random variables, but it can also produce new results for the sum of random variables with heavier tails. We show that the concentration inequalities we obtain are sharp enough to offer large deviation results for the sums of independent random variables as well. Our analyses which are based on standard truncation arguments simplify, unify and generalize the existing results on the concentration and large deviation of heavy-tailed random variables.