Some Characterizations and Properties of COM-Poisson Random Variables
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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_ν of the discrete random variable X. The probability mass function of X_ν has a link to the Rényi entropy and Tsallis entropy of order ν 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 (ν 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.
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