Abstract: The notion of -stability of van Harn et al. [10] (see also Steutel and van Harn
[20]) and the related concept of -semistability are intimately connected with
continuous-time branching processes. -stable and -semistable distributions play also a
significant role in the theory of integer-valued (semi-)self-similar processes and have
arisen as stationary solutions of integer-valued autoregressive processes. The aim of
this article is twofold. Firstly, we provide several new characterizations of uni-
variate -stable and -semistable distributions. Secondly, we propose a systematic study
of -stability and -semistability for distributions on the -dimensional lattice
.