A note on mixed Poisson distributions

TL;DR

This note analyzes mixed Poisson distributions with a random mixing variable $X$ and scale $\rho$, focusing on asymptotic limit laws as $\rho \to \infty$ via factorial moments. It proves that $Y/\rho$ converges to the mixing variable $X$ in distribution and in probability, and establishes a central limit theorem for $\sqrt{\rho}(\tfrac{Y}{\rho}-X)$ with limit distribution $\mathcal{N}(0,X)$, along with moment convergence. The work also provides explicit raw-moment expressions, discusses degenerate and zero-mass cases, and extends to multivariate mixed Poisson distributions. These results strengthen previous limit theorems for combinatorial structures and emphasize the role of using the same mixing variable for centering and scaling, with potential applications to processes such as the Chinese restaurant process.

Abstract

In this note we discuss additional properties of mixed Poisson distributions. We discuss the convergence of mixed Poisson distributions to its mixing distribution for the scaling parameter tending to infinity. Moreover, we obtain a central limit theorem after centering by its mixing random variable, together with moment convergence.

Theorems & Definitions (10)

• Definition 1
• Lemma 1: Factorial moments and limit laws of mixed Poisson type KuPa2014
• Theorem 1
• Remark 1
• Remark 2: Shifting of $Y=\mathop{\mathrm{\text{MPo}}}\nolimits(\rho X)$
• Remark 3: Point mass at zero
• Theorem 2
• proof : Proof of Theorem \ref{['the1']}
• proof : Proof of Theorem \ref{['the2']}
• Definition 2