AISM 52, 790-799

On geometric infinite divisibility and stability

Emad-Eldin A.A. Aly1 and Nadjib Bouzar2

1Department of Statistics and O.R., Kuwait University, P.O.B. 5969, Safat 13060, Kuwait
2Department of Mathematics, University of Indianapolis, Indianapolis, IN 46227, U.S.A.

(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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