Asymptotic formulas for products of Poisson distributions

Abstract

In this paper, we study the asymptotic behaviour of the product tail probability $\mathbb{P}({\xi }_1\cdots {\xi }_N⩾n),$ where $\{{\xi }_1,\dots ,{\xi }_N\}$ is a finite collection of independent Poisson random variables with positive parameters ${\lambda }_1,\dots ,{\lambda }_N$. We derive a refined Laplace-type asymptotic formula for the tail probability, based on Stirling's logarithmic approximation, a constrained saddle-point method, the Lambert function, and a careful evaluation of the constrained Gaussian prefactor. This yields an explicit approximation with an $O(\log n)$ remainder term in the exponent.