The density ratio of Poisson binomial versus Poisson distributions

Abstract

Let $b(x)$ be the probability that a sum of independent Bernoulli random variables with parameters $p_1,p_2,p_3,\dots \in [0,1)$ 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 $\lambda$.

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