Some Characterizations and Properties of COM-Poisson Random Variables

TL;DR

This paper explores new characterizations and properties of COM-Poisson random variables, including extensions of classical characterizations, entropy connections, and a novel Poisson distribution characterization based on addition properties.

Contribution

It extends classical Poisson characterizations to COM-Poisson variables and introduces a new Poisson characterization based on their non-closure under addition.

Findings

Extended Moran-Chatterji and Rao-Rubin characterizations to COM-Poisson

Linked COM-type discrete entropy to Rényi and Tsallis entropies

Identified non-closure under addition for COM-Poisson variables with ν ≠ 1

Abstract

This paper introduces some new characterizations of COM-Poisson random variables. First, it extends Moran-Chatterji characterization and generalizes Rao-Rubin characterization of Poisson distribution to COM-Poisson distribution. Then, it defines the COM-type discrete r.v. ${X_\nu }$ of the discrete random variable $X$. The probability mass function of ${X_\nu }$ has a link to the R\'enyi entropy and Tsallis entropy of order $\nu $ of $X$. And then we can get the characterization of Stam inequality for COM-type discrete version Fisher information. By using the recurrence formula, the property that COM-Poisson random variables ($\nu \ne 1$) is not closed under addition are obtained. Finally, under the property of "not closed under addition" of COM-Poisson random variables, a new characterization of Poisson distribution is found.