Summary
An intrinsic definition of sup self-decomposable random vectors is given. It is proved that they are precisely the limits in distribution of certain normalized partial maxima of sequences of independent random vectors. The main further result is a representation of sup self-decomposable random vectors as functions of Poisson processes, which is the analogue of Wolfe's (1982) representation of additively self-decomposable random variables.
References
Balkema, A., Resnick, S.: Max-infinite divisibility. J. Appl. Probab. 14, 309–319 (1977)
Berg, C., Reus Christensen, J., Ressel, P.: Prositive definite functions on Abelian semigroups. Math. Ann. 223, 253–274 (1976)
De Haan, L., Resnick, S.: Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 40, 317–337 (1977)
Deheuvels, P.: Caractérisation des lois extrêmes multivariées et de la convergence des types extrêmes. Publ. Inst. Univ. Paris 23, 1–36 (1978)
Deheuvels, P.: The decomposition of infinite order and extreme multivariate distributions. In: Asymptotic theory of statistical tests and estimation — In honor of Wassily Höffding (ed. I.M. Chakravarti). New York: Academic Press 1980
Galambos, J.: The asymptotic theory of extreme order statistics. New York: Wiley 1978
Gierz, G., Hofmann, K., Keimel, K., Lawson, J., Mislove, M., Scott, D.: A compendium of continuous lattices. Berlin Heidelberg New York: Springer 1980
Jurek, Z., Vervaat, W.: An integral representation for selfdecomposable Banach space valued random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb. 62, 247–262 (1983)
Kallenberg, O.: Random measures. Berlin: Akademie Verlag and London: Academic Press 1976
Laha, R., Rohatgi, V.: Probability theory. New York: Wiley 1979
Loève, M.: Probability theory, vol. I, 4th edition. Berlin Heidelberg New York: Springer 1977
Lukacs, E.: Characteristics functions. London: Griffin 1960
Matheron, G.: Random sets and integral geometry. New York: Wiley 1975
Mejzler, D.: On the limit distribution of the maximal term of a variational series (in Ukrainian). Dopovidi Akad. Nauk Ukrain, SSR 1, 3–10 (1950)
Mejzler, D.: The study of the limit laws for the variational series (in Russian). Trudy Inst. Mat. Akad. Nauk Uzbek. SSR 10, 96–105 (1953)
Mejzler, D.: On the problem of the limit distribution for the maximal term of a variational series (in Russian). L'vov Politechn. Inst. Naucn. Zp. (Fiz-Mat.) 38, 90–109 (1956)
Norberg, T.: Convergence and existence of random set distributions. Ann. Probab. 12, 726–732 (1984)
Norberg, T.: Random capacities and their distributions. Report 1984-03, Dept. of Math., University of Göteborg 1984
Roberts, A., Varberg, D.: Convex functions. New York: Academic Press 1973
Wolfe, S.: On a continuous analogue of the stochastic difference equation 33-1 Stoch. Proc. Appl. 12, 301–312 (1982)
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The present paper grew out of a Master's Thesis under supervision of Wim Vervaat. Support was provided by the Netherlands Organization for the Advancement of Pure Research ZWO via the Mathematical Centre Foundation SMC (project 10-62-07)
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Gerritse, G. Supremum self-decomposable random vectors. Probab. Th. Rel. Fields 72, 17–33 (1986). https://doi.org/10.1007/BF00343894
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DOI: https://doi.org/10.1007/BF00343894
Keywords
- Stochastic Process
- Probability Theory
- Statistical Theory
- Random Vector
- Poisson Process