Abstract
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.
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Aly, EE.A.A., Bouzar, N. On Geometric Infinite Divisibility and Stability. Annals of the Institute of Statistical Mathematics 52, 790–799 (2000). https://doi.org/10.1023/A:1017589613321
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DOI: https://doi.org/10.1023/A:1017589613321