Sharp large deviation results for sums of independent random variables

TL;DR

This paper derives sharp large deviation bounds for sums of independent variables under Bernstein's condition, improving existing inequalities and providing precise asymptotic expansions similar to classical results.

Contribution

It introduces new tight bounds for large deviations, enhances Talagrand's inequality with matching lower bounds, and refines inequalities by Pinelis, advancing the theoretical understanding of tail probabilities.

Findings

Bounds are close to Gaussian tail in certain cases

Improves Bennett and Hoeffding inequalities with additional factors

Provides large deviation expansions akin to classical results

Abstract

We show sharp bounds for probabilities of large deviations for sums of independent random variables satisfying Bernstein's condition. One such bound is very close to the tail of the standard Gaussian law in certain case; other bounds improve the inequalities of Bennett and Hoeffding by adding missing factors in the spirit of Talagrand (1995). We also complete Talagrand's inequality by giving a lower bound of the same form, leading to an equality. As a consequence, we obtain large deviation expansions similar to those of Cram\'{e}r (1938), Bahadur-Rao (1960) and Sakhanenko (1991). We also show that our bound can be used to improve a recent inequality of Pinelis (2014).