Characterizations of discrete compound Poisson distributions

The aim of this paper is to give some new characterizations of discrete compound Poisson distributions. Firstly, we give a characterization by the Levy-Khinchine formula of infinitely divisible distributions under some conditions. The second characterization need to present by row sum of random triangular arrays converges in distribution. And we give an application in probabilistic number theory, the strongly additive function converging to a discrete compound Poisson in distribution. The next characterization, is an extension of Watanabe's theorem of characterization of homogeneous Poisson process. The last characterization will be illustrated by waiting time distributions, especially the matrix-exponential representation.