Characterization of count data distributions involving additivity and binomial subsampling

In this paper we characterize all the r-parameter families of count distributions (satisfying mild conditions) that are closed under addition and under binomial subsampling. Surprisingly, few families satisfy both properties and the resulting models consist of the rth-order univariate Hermite distributions. Among these, we find the Poisson $(r=1)$ and the ordinary Hermite distributions $(r=2)$.