Geometric stable and semistable distributions on Z+d

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 proces of order p with a geometric F-stable marginal distribution.

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Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).