Asymptotic formulas for products of Poisson distributions

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.