On Geometric Infinite Divisibility and Stability

The purpose of this paper is to study geometric infinite divisibility and geometric stability of distributions with support in Z ₊ and R ₊. Several new characterizations are obtained. We prove in particular that compound-geometric (resp. compound-exponential) distributions form the class of geometrically infinitely divisible distributions on Z ₊ (resp. R ₊). These distributions are shown to arise as the only solutions to a stability equation. We also establish that the Mittag-Leffler distributions characterize geometric stability. Related stationary autoregressive processes of order one (AR(1)) are constructed. Importantly, we will use Poisson mixtures to deduce results for distributions on R ₊ from those for their Z ₊-counterparts.