On Geometric Infinite Divisibility and Stability

Authors: Aly E-E.A.A.¹; Bouzar N.²

Source: Annals of the Institute of Statistical Mathematics, Volume 52, Number 4, December 2000 , pp. 790-799(10)

Publisher: Springer

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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.

Keywords: Geometric infinite divisibility; geometric stability; compound-geometric; compound-exponential; Mittag-Leffler; Poisson mixtures; Lévy process

Language: English

Document Type: Regular paper

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

Publication date: 2000-12-01