Asymptotic formulas for products of Poisson distributions

TL;DR

This paper derives a refined asymptotic formula for the tail probability of the product of independent Poisson variables, providing explicit approximations with controlled error terms.

Contribution

It introduces a new asymptotic formula for the product tail probability of Poisson variables using advanced analytical techniques.

Findings

Derived a Laplace-type asymptotic formula with an explicit $O(\log n)$ error term.

Utilized saddle-point method and Lambert function for precise tail probability approximation.

Provides a detailed evaluation of the Gaussian prefactor in the asymptotic analysis.

Abstract

In this paper, we study the asymptotic behaviour of the product tail probability $ \mathbb{P}(\xi_1\cdots\xi_N \geqslant n), $ where $\{\xi_1,\ldots,\xi_N\}$ is a finite collection of independent Poisson random variables with positive parameters $\lambda_1,\ldots,\lambda_N$. 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.