Discrete broadly stable distributions

A random variable X is strictly stable if a sum of independent copies of X has the same distribution as X up to scaling, and is stable (in the broad sense) if the sum has the same distribution as X up to both scaling and shifting. Steutel and Van Harn studied 'discrete strict stability' for random variables taking values in the non-negative integers, replacing scaling by the thinning operation. We extend this to 'discrete stability' (in the broad sense), where shifting is replaced by the addition of an independent Poisson random variable. The discrete stable distributions are 'Poisson--delayed Sibuya' distributions, which include the Poisson and Hermite distributions.