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

Characterization of count data distributions involving additivity and binomial subsampling