An asymptotically Gaussian bound on the Rademacher tails

TL;DR

This paper establishes an explicit, asymptotically optimal Gaussian bound on the tail probabilities of Rademacher sums, confirming a long-standing conjecture and extending to applications in bounded sums and statistical tests.

Contribution

It provides a new, best-possible upper bound on Rademacher tail probabilities that aligns asymptotically with the normal distribution, confirming Efron's conjecture.

Findings

The bound is asymptotically equivalent to the normal tail probability.

It confirms the conjecture by Efron regarding Rademacher sums.

Applications include bounds for sums of bounded independent variables and the Student test.

Abstract

An explicit upper bound on the tail probabilities for the normalized Rademacher sums is given. This bound, which is best possible in a certain sense, is asymptotically equivalent to the corresponding tail probability of the standard normal distribution, thus affirming a longstanding conjecture by Efron. Applications to sums of general centered uniformly bounded independent random variables and to the Student test are presented.