Abstract
The multiplications of van Harn et al. (1982, Z. Wahrsch. Verw. Gebiete, 61, 97–118) are used to generalize the definition of α-monotonicity of Olshen and Savage (1970, J. Appl. Probab., 7, 21–34) and Steutel (1988, Statist. Neerlandica, 42, 137–140) for distributions with support in Z + and R +. Several characterizations are offered and a convolution property is established. Some relevant stability equations are solved and a relationship with the important concept of self-decomposability is noted. Poisson mixtures are used to deduce results for the R +-case from those for the Z +-case.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abouammoh, A. M. (1987). On discrete α-unimodality, Statist. Neerlandica, 41, 239–244.
Alamatsaz, M. H. (1993). On discrete α-unimodal distributions, Statist. Neerlandica, 47, 245–252.
Alzaid, A. A. and Al-Osh, M. A. (1990). Some results on discrete α-monotonicity, Statist. Neerlandica, 44, 29–33.
Athreya, K. B. and Ney, P. E. (1972). Branching Processes, Springer, Berlin-Heidelberg-New York.
Bouzar, N. (1999). On geometric stability and Poisson mixtures, Illinois J. Math., 43, 520–527.
Devroye, L. (1993). A triptych of discrete distributions related to the stable law, Statist. Probab. Lett., 18, 349–351.
Hansen, B. G. (1989). Self-decomposable distributions and branching processes, Memoir COSOR 89–06, Eindhoven University of Technology, Eindhoven, The Netherlands.
Hansen, B. G. (1990). Monotonicity properties of infinitely divisible distributions, Centrum voor Wiskunde en Informatica, Tract 69, Amsterdam, The Netherlands.
Huang, W. and Chen, L. (1989). Note on a characterization of gamma distributions, Statist. Probab. Lett., 8, 485–487.
Knopp, K. (1990). Theory and Applications of Infinite Series, Dover, New York.
Olshen, R. A. and Savage, L. J. (1970). A generalized unimodality, J. Appl. Prob., 7, 21–34.
Steutel, F. W. (1988). Note on discrete α-unimodality, Statist. Neerlandica, 42, 137–140.
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. Appl., 45, 209–230.
van Harn, K., Steutel, F. W. and Vervaat, W. (1982). Self-decomposable discrete distributions and branching processes, Z. Wahrsch. Verw. Gebiete, 61, 97–118.
Wu, F. and Dharmadhikari, S. (1999). Convolutions of α-monotone distributions, Statist. Neerlandica, 53, 247–250.
Author information
Authors and Affiliations
About this article
Cite this article
Aly, EE.A.A., Bouzar, N. A Notion of α-Monotonicity with Generalized Multiplications. Annals of the Institute of Statistical Mathematics 54, 125–137 (2002). https://doi.org/10.1023/A:1016121906623
Issue Date:
DOI: https://doi.org/10.1023/A:1016121906623