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On the semi-Mittag-Leffler distributions

Lin Gwo Dong, Hu Chin-Yuan

Probability and Mathematical Statistics

2022

Vol. 42, Fasc. 1

81--9

The semi-Mittag-Leffler (SML) distribution arises as the marginal of a stationary Markovian process, and is a generalization of the well-known Mittag-Leffler (ML) or positive Linnik distribution. Unlike the ML distribution, which has been well established, few properties of the SML distribution are discussed in the literature. In this paper, we derive some more characterizations of the SML and related distributions. By using stochastic inequalities, we further extend some characterizations, including Pitman and Yor’s (2003) result about the hyperbolic sine distribution.

Geometric stable and semistable distributions on Z+d

Bouzar N.

Probability and Mathematical Statistics

2015

Vol. 35, Fasc. 2

223--245

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.