The Kearns--Saul inequality for Bernoulli and Poisson-binomial distributions

TL;DR

This paper provides a rigorous proof of the Kearns--Saul inequality for generalized Bernoulli variables, extends it to Poisson-binomial distributions, and characterizes parameter conditions for sums of Bernoulli variables.

Contribution

It offers a direct proof of the Kearns--Saul inequality and extends it to more general distributions, including sums of Bernoulli variables.

Findings

Rigorous proof of the Kearns--Saul inequality.

Extension of the inequality to Poisson-binomial distributions.

Characterization of parameters for sums of Bernoulli variables.

Abstract

We give a direct rigorous proof of the Kearns--Saul inequality which bounds the Laplace transform of a generalised Bernoulli random variable. We extend the arguments to generalised Poisson-binomial distributions and characterise the set of parameters such that an analogous inequality holds for the sum of two generalised Bernoulli random variables.