Tight Bounds on the Binomial CDF, and the Minimum of i.i.d Binomials, in terms of KL-Divergence

TL;DR

This paper derives finite sample, asymptotically tight bounds on binomial tail probabilities and the minimum of i.i.d. binomials using KL-divergence, based on Sanov's theorem.

Contribution

It introduces new finite sample bounds on binomial tail probabilities and the minimum of i.i.d. binomials expressed via KL-divergence, extending Sanov's theorem.

Findings

Bounds are finite sample and asymptotically tight.

Bounds are expressed in terms of KL-divergence.

Applicable to high probability estimates of binomial minima.

Abstract

We provide finite sample upper and lower bounds on the Binomial tail probability which are a direct application of Sanov's theorem. We then use these to obtain high probability upper and lower bounds on the minimum of i.i.d. Binomial random variables. Both bounds are finite sample, asymptotically tight, and expressed in terms of the KL-divergence.