On geometric infinite divisibility and stability (Q5936215)

The paper studies geometric infinite divisibility and geometric stability of distributions with support on the nonnegative integers \(\mathbb Z_+\) and the nonnegative reals \(\mathbb R_+\), respectively. Several new characterizations are obtained. It is shown that compound-geometric (resp. compound-exponential) distributions form the class of geometrically infinitely divisible distributions on \(\mathbb Z_+\) (resp. \(\mathbb R_+\)). Furthermore it is shown that the Mittag-Leffler distributions characterize geometric stability. The authors use Poisson mixtures to deduce results for distributions on \(\mathbb R_+\) from those of their \(\mathbb Z_+\)-counterparts.