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Contents of PMS, Vol. 33, Fasc. 1,
pages 149 - 174
 

CHARACTERIZATIONS OF F -STABLE AND F -SEMISTABLE DISTRIBUTIONS

Nadjib Bouzar

Abstract: The notion of F -stability of van Harn et al. [10] (see also Steutel and van Harn [20]) and the related concept of F -semistability are intimately connected with continuous-time branching processes. F -stable and F -semistable distributions play also a significant role in the theory of integer-valued (semi-)self-similar processes and have arisen as stationary solutions of integer-valued autoregressive processes. The aim of this article is twofold. Firstly, we provide several new characterizations of uni- variate F -stable and F -semistable distributions. Secondly, we propose a systematic study of F -stability and F -semistability for distributions on the d -dimensional lattice Zd
  +  .

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

Keywords and phrases: Semigroup, infinite divisibility, branching processes, probability generating functions, discrete multivariate distributions, the Lau–Rao theorem.

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