AISM 52, 790-799
(Received March 18, 1998; revised May 4, 1999)
Abstract. The purpose of this paper is to study geometric infinite divisibility and geometric stability of distributions with support in $\bz_+$ and $\br_+$. 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 $\bz_+$ (resp. $\br_+$). 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 $\br_+$ from those for their $\bz_+$-counterparts.
Key words and phrases: Geometric infinite divisibility, geometric stability, compound-geometric, compound-exponential, Mittag-Leffler, Poisson mixtures, Lévy process.
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