ON GEOMETRIC INFINITE DIVISIBILITY AND STABILITY

EMAD-ELDIN A. A. ALY^{1} AND NADJIB BOUZAR^{2}

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 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.

Key words and phrases: Geometric infinite divisibility, geometric stability, compound-geometric, compound-exponential, Mittag-Leffler, Poisson mixtures, Lévy process.