Nearly tight universal bounds for the binomial tail probabilities

TL;DR

This paper presents simple, nearly tight bounds for binomial tail probabilities that outperform traditional bounds like Chernoff, applicable across all parameter regimes with strong asymptotic tightness and a surprising connection to Ramanujan's equation.

Contribution

The authors introduce new bounds for binomial tail probabilities that are easy to compute, tight within a constant factor, and improve upon existing Chernoff bounds across all regimes.

Findings

Bounds are tight within a factor of 89/44.

Asymptotically tight in large and moderate deviation regimes.

Bounds outperform Chernoff and reverse Chernoff bounds.

Abstract

We derive simple but nearly tight upper and lower bounds for the binomial lower tail probability (with straightforward generalization to the upper tail probability) that apply to the whole parameter regime. These bounds are easy to compute and are tight within a constant factor of $89/44$. Moreover, they are asymptotically tight in the regimes of large deviation and moderate deviation. By virtue of a surprising connection with Ramanujan's equation, we also provide strong evidences suggesting that the lower bound is tight within a factor of $1.26434$. It may even be regarded as the natural lower bound, given its simplicity and appealing properties. Our bounds significantly outperform the familiar Chernoff bound and reverse Chernoff bounds known in the literature and may find applications in various research areas.