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Asymptotic Tail of the Product of Independent Poisson Random Variables

Džiugas Chvoinikov, Jonas Šiaulys

Abstract

This paper studies the asymptotic tail behaviour of products of independent Poisson random variables. Let \[ Z_m=\prod_{j=1}^m X_j, \] where $X_1,\dots,X_m$ are independent Poisson random variables. We derive a Laplace-type asymptotic approximation for \[ P(Z_m \ge n), \qquad n\to\infty, \] whose relative error tends to zero. The analysis is based on Stirling's logarithmic approximation, a constrained saddle-point method, the Lambert $W$ function, and a careful evaluation of the associated Gaussian prefactor. These tools yield an explicit asymptotic description of the tail probability of the product. For clarity of exposition, we first treat the case $m=2$, which illustrates the main ideas in a simpler setting, and then extend the argument to the general product of $m$ independent Poisson random variables.

Asymptotic Tail of the Product of Independent Poisson Random Variables

Abstract

This paper studies the asymptotic tail behaviour of products of independent Poisson random variables. Let where are independent Poisson random variables. We derive a Laplace-type asymptotic approximation for whose relative error tends to zero. The analysis is based on Stirling's logarithmic approximation, a constrained saddle-point method, the Lambert function, and a careful evaluation of the associated Gaussian prefactor. These tools yield an explicit asymptotic description of the tail probability of the product. For clarity of exposition, we first treat the case , which illustrates the main ideas in a simpler setting, and then extend the argument to the general product of independent Poisson random variables.
Paper Structure (34 sections, 4 theorems, 236 equations, 3 figures)

This paper contains 34 sections, 4 theorems, 236 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Products of Random Variables
  3. Previous Research
  4. Main Results
  5. Tools and Auxiliary Results
  6. Stirling's Logarithmic Approximation
  7. Lagrange Multiplier Setup for the Constraint kℓ = n
  8. Lambert--W Representation of the Saddle Point
  9. Constrained Hessian for a Single Constraint
  10. Multidimensional Laplace Approximation
  11. Proof of the Main Result
  12. Decomposition into three regimes
  13. Comparison with the upper bounds for $R_1$ and $R_2$.
  14. Saddle-point setup on the main regime $R_3$
  15. Solution of the Lagrange System (Lambert W Form)
  16. ...and 19 more sections

Key Result

Theorem 2.1

Let $X \sim \mathrm{Pois}(\lambda_1)$ and $Y \sim \mathrm{Pois}(\lambda_2)$ be independent Poisson random variables with parameters $\lambda_1,\lambda_2 > 0$. For $n \ge 1$ define Let $(k_n^*,\ell_n^*)$ be the unique solution of for some real $\alpha_n$. Then, as $n \to \infty$,

Figures (3)

  • Figure 1: Logarithmic comparison of the exact tail probability $p_n=\mathbb{P}(X_1X_2\ge n)$ and the Laplace approximation from Theorem \ref{['thm:main']}, evaluated at the numerical saddle point. The parameters are $\lambda_1=2$ and $\lambda_2=3$.
  • Figure 2: Comparison of the exact log–tail and the first three terms of the asymptotic expansion \ref{['eq:L123']} for $\lambda_1=2$, $\lambda_2=3$.
  • Figure 3: Log–tail probabilities $\log p_n^{(m)}$ for $m=2,3,4,5$ (equal $\lambda_i$). Increasing the dimension systematically slows the decay rate, in agreement with $\log p_n^{(m)}\sim -n^{1/m}\log n$.

Theorems & Definitions (6)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1: Multidimensional Laplace Method wong2001
  • Theorem 5.1
  • proof
  • Remark 5.2: Practical two--term approximation