Analogues are proposed for the concepts of self-decomposability and stability for distributions on the nonnegative integers. It turns out that these "discrete self-decomposable" and "discrete stable" distributions have properties that are quite similar to those of their continuous counterparts.
"Discrete Analogues of Self-Decomposability and Stability." Ann. Probab. 7 (5) 893 - 899, October, 1979. https://doi.org/10.1214/aop/1176994950