Abstract: The aim of this article is to study geometric -semistable and geometric -stable
distributions on the -dimensional lattice . We obtain several properties for these
distributions, including characterizations in terms of their probability generating functions.
We describe a relation between geometric -semistability and geometric -stability and
their counterparts on 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 . As an application,
we derive the marginal distribution of the innovation sequence of a -valued
stationary autoregressive process of order with a geometric -stable marginal
distribution.