When Is the Conway-Maxwell-Poisson Distribution Infinitely Divisible?

TL;DR

This paper investigates the infinite divisibility of the Conway-Maxwell-Poisson distribution, establishing that it is only infinitely divisible when it reduces to a Poisson or geometric distribution, thus limiting its theoretical role in limit theorems.

Contribution

It provides a definitive characterization of when the CMP distribution is infinitely divisible, clarifying its theoretical limitations.

Findings

CMP is infinitely divisible only as Poisson or geometric

Clarifies the theoretical foundation of CMP in limit theory

Limits the use of CMP as a natural law of small numbers

Abstract

An essential character for a distribution to play a central role in the limit theory is infinite divisibility. In this note, we prove that the Conway-Maxwell-Poisson (CMP) distribution is infinitely divisible iff it is the Poisson or geometric distribution. This explains that, despite its applications in a wide range of fields, there is no theoretical foundation for the CMP distribution to be a natural candidate for the law of small numbers.