Sharp large deviation probabilities for sums of independent bounded random variables

TL;DR

This paper derives optimal tail probability bounds for sums of independent bounded variables, providing a sharp asymptotic expansion for large deviations that enhances previous inequalities and aligns with classical large deviation results.

Contribution

It introduces a one-term asymptotic expansion for large deviations of bounded sums, improving Talagrand's bounds and connecting to classical large deviation theories.

Findings

Established optimal tail probability inequalities.

Provided a sharp asymptotic expansion for large deviations.

Extended classical large deviation results to bounded variables.

Abstract

We obtain some optimal inequalities on tail probabilities for sums of independent bounded random variables. Our main result completes an upper bound on tail probabilities due to Talagrand by giving a one-term asymptotic expansion for large deviations. This result can also be regarded as sharp large deviations of types of Cram\'er and Bahadur-Ranga Rao.