Geometric stable and semistable distributions on Z+^d

Abstrakt

The aim of this article is to study geometric F-semistable and geometric F-stable distributions on the d-dimensional lattice Z₊^d. 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 R₊^d 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 Z₊^d. As an application, we derive the marginal distribution of the innovation sequence of a Z₊^d-valued stationary autoregressive process of order p with a geometric F-stable marginal distribution.