Abstract
In this paper the possible nondegenerated limit distributions for the n-fold mapping of a given probability distribution are considered. If the mapping used for the iteration procedure is a probability generating function of a positive integer-valued random variable then the results can be applied to the max-stability of distributions of random variables with random sample size.
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References
Aczél, J. (1961). Vorlesungen über Funktionalgleichungen und ihre Anwendungen, Birkhäuser, Basel.
Baringhaus, L. (1980). Eine simultane Charakterisierung der geometrischen Verteilung und der logistischen Verteilung, Metrika, 27, 243–253.
Brücks, G. (1993). Verteilungseigenschaften von Ordnungsstatistiken bei zufälligem Stichprobenumfang, Ph.D. Thesis, Aachen University of Technology, Germany.
Bunge, J. (1993). Some stability classes for random numbers of random vectors, Comm. Statist. Stochastic Models, 9(2), 247–253.
Gensler, H. (1992). Stabilitätseigenschaften von Verteilungsfunktionen, Master Thesis, Aachen University of Technology.
Gnedenko, B. V. (1982). On some stability theorems, Stability Problems for Stochastic Models, Proc. 6th Seminar, Moscow (eds. V. V. Kalashnikov and V. M. Zolotarev), Lecture Notes in Math., 982, 24–31, Springer, Berlin.
Gnedenko, B. V. (1983). On limit theorems for a random number of random variables. Probability Theory and Mathematical Statistics, Fourth UDSSR-Japan Symposium, Lecture Notes in Math., 1021, 167–176, Springer, Berlin.
Kremer, E. (1983). Distribution-free upper bounds on the premiums of the LCR and ECOMOR treaties, Insurance Math. Econom., 2, 209–213.
Kruglov, V. M. and Korolev, V. (1990). Limit Theorems for Random Sums, Moscow University Press.
Mittnik, S. and Rachev, S. T. (1991). Alternative multivariate distributions and their applications to financial modeling, Stable Processes and Related Topics (eds. S. Cambanis, G. Samorodnitsky and M. S. Taqqu), 107–119, Birkhäuser, Boston.
Mittnik, S. and Rachev, S. T. (1993). Modeling asset returns with alternative stable distributions, Econometric Rev., 12(3), 261–330.
Rachev, S. T. (1991). Probability Metrics and the Stability of Stochastic Models, Wiley, New York.
Rachev, S. T. and Resnick, S. (1991). Max-geometric infinite divisibility and stability, Comm. Statist. Stochastic Models, 7, 191–218.
Rachev, S. T. and Samorodnitsky, G. (1992). Geometric stable distributions in Banach Spaces, Tech. Report, Cornell University.
Rachev, S. T. and Sen Gupta, A. (1992). Geometric stable distributions and Laplace-Weibull mixtures, Statist. Decisions, 10, 251–271.
Voorn, W. J. (1987). Characterization of the logistic and loglogistic distributions by extreme values related stability with random sample size, J. Appl. Probab., 24, 838–851.
Voorn, W. J. (1989). Stability of extremes with random sample size, J. Appl. Probab., 27, 734–743.
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Rauhut, B. Iterated probability distributions and extremes with random sample size. Ann Inst Stat Math 48, 145–155 (1996). https://doi.org/10.1007/BF00049295
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DOI: https://doi.org/10.1007/BF00049295