Some characterizations and properties of COM-Poisson random variables

Starting with a literature review for theoretical properties of COM-Poisson distributions, this paper proposes some new characterizations of COM-Poisson random variables. First, we extend the Moran-Chatterji characterization and generalize the Rao-Rubin characterization of Poisson distribution to COM-Poisson distribution. Then, we define the COM-type discrete r.v. of the discrete random variable X. The probability mass function of has a link to the Renyi 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 () is not closed under addition is obtained. Finally, under the property of "not closed under addition" of COM-Poisson random variables, a new characterization of Poisson distribution is found.