Abstract
The purpose of this paper is to study geometric infinite divisibility and geometric stability of distributions with support in Z + and R +. Several new characterizations are obtained. We prove in particular that compound-geometric (resp. compound-exponential) distributions form the class of geometrically infinitely divisible distributions on Z + (resp. R +). These distributions are shown to arise as the only solutions to a stability equation. We also establish that the Mittag-Leffler distributions characterize geometric stability. Related stationary autoregressive processes of order one (AR(1)) are constructed. Importantly, we will use Poisson mixtures to deduce results for distributions on R + from those for their Z +-counterparts.
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Alzaid, A. A. and Al-Osh, M. A. (1990). Some results on discrete α-monotonicity, Statist. Neerlandica, 44, 29–33.
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd. ed., Wiley, New York.
Fujita, Y. (1993). A generalization of the results of Pillai, Ann. Inst. Statist. Math., 45, 361–365.
Gaver, D. P. and Lewis, P. A. W. (1980). First-order autoregressive gamma sequences and point processes, Advances in Applied Probability, 12, 727–745.
Gnedenko, B. V. and Korolev, V. Y. (1996). Random Summation: Limit Theorems and Applications, CRC Press, Boca Raton.
Klebanov, L. B., Maniya, G. M. and Melamed, I. A. (1984). A problem of Zolotarev and analogs of infinitely divisible and stable distributions in a scheme for summing a random number of random variables, Theory Probab. Appl., 29, 791–794.
Lawrance, A. J. (1982). The innovation distribution of a gamma distributed autorgressive process, Scan. J. Statist., 9, 234–236.
McKenzie, E. (1986). Autoregressive-moving average processes with negative binomial and geometric marginal distributions, Advances in Applied Probability, 18, 679–705.
McKenzie, E. (1987). Innovation distributions for gamma and negative binomial autoregressions, Scan. J. Statist., 14, 79–85.
Pillai, R. N. (1990). On Mittag-Leffler functions and related distributions, Ann. Inst. Statist. Math., 42, 157–161.
Pillai, R. N. and Jayakumar, K. (1995) Discrete Mittag-Leffler distributions, Statist. Probab. Lett., 23, 271–274.
Steutel, F. W. and van Harn, K. (1979). Discrete analogues of self-decomposability and stability, Ann. Probab., 7, 893–899.
van Harn, K. and Steutel, F. W. (1993). Stability equations for processes with stationary independent increments using branching processes and Poisson mixtures. Stochastic Processes and Their Applications, 45, 209–230.
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Aly, EE.A.A., Bouzar, N. On Geometric Infinite Divisibility and Stability. Annals of the Institute of Statistical Mathematics 52, 790–799 (2000). https://doi.org/10.1023/A:1017589613321
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DOI: https://doi.org/10.1023/A:1017589613321