The density ratio of Poisson binomial versus Poisson distributions

TL;DR

This paper investigates the ratio between the probability mass function of a Poisson binomial distribution and a Poisson distribution, establishing bounds for their maximal ratio.

Contribution

It provides new inequalities for the maximal ratio of Poisson binomial to Poisson probabilities, enhancing understanding of their relationship.

Findings

Derived bounds for the maximal ratio of b(x) to π_λ(x)

Established inequalities for the density ratio

Improved approximation techniques for Poisson binomial distributions

Abstract

Let $b(x)$ be the probability that a sum of independent Bernoulli random variables with parameters $p_1, p_2, p_3, \ldots \in [0,1)$ equals $x$, where $\lambda := p_1 + p_2 + p_3 + \cdots$ is finite. We prove two inequalities for the maximal ratio $b(x)/\pi_\lambda(x)$, where $\pi_\lambda$ is the weight function of the Poisson distribution with parameter $\lambda$.