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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 35, Fasc. 2,
pages 223 - 245
 

GEOMETRIC STABLE AND SEMISTABLE DISTRIBUTIONS ON Zd+

Nadjib Bouzar

Abstract: The aim of this article is to study geometric F -semistable and geometric F -stable distributions on the d -dimensional lattice Zd + . We obtain several properties for these distributions, including characterizations in terms of their probability generating functions. We describe a relation between geometric F -semistability and geometric F -stability and their counterparts on Rd + and, as a consequence, we derive some mixture representations and construct some examples. We establish limit theorems and discuss the related concepts of complete and partial geometric attraction for distributions on Zd + . As an application, we derive the marginal distribution of the innovation sequence of a Zd + -valued stationary autoregressive process of order p with a geometric F -stable marginal distribution.

2000 AMS Mathematics Subject Classification: Primary: 60E07; Secondary: 62E10.

Keywords and phrases: Semigroup, geometric infinite divisibility, branching processes, mixture representation, domain of geometric attraction.

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