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Supremum self-decomposable random vectors
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  • Published: April 1986

Supremum self-decomposable random vectors

  • Gerard Gerritse1 

Probability Theory and Related Fields volume 72, pages 17–33 (1986)Cite this article

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.

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References

  1. Balkema, A., Resnick, S.: Max-infinite divisibility. J. Appl. Probab. 14, 309–319 (1977)

    Google Scholar 

  2. Berg, C., Reus Christensen, J., Ressel, P.: Prositive definite functions on Abelian semigroups. Math. Ann. 223, 253–274 (1976)

    Google Scholar 

  3. De Haan, L., Resnick, S.: Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 40, 317–337 (1977)

    Google Scholar 

  4. Deheuvels, P.: Caractérisation des lois extrêmes multivariées et de la convergence des types extrêmes. Publ. Inst. Univ. Paris 23, 1–36 (1978)

    Google Scholar 

  5. 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

    Google Scholar 

  6. Galambos, J.: The asymptotic theory of extreme order statistics. New York: Wiley 1978

    Google Scholar 

  7. Gierz, G., Hofmann, K., Keimel, K., Lawson, J., Mislove, M., Scott, D.: A compendium of continuous lattices. Berlin Heidelberg New York: Springer 1980

    Google Scholar 

  8. Jurek, Z., Vervaat, W.: An integral representation for selfdecomposable Banach space valued random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb. 62, 247–262 (1983)

    Google Scholar 

  9. Kallenberg, O.: Random measures. Berlin: Akademie Verlag and London: Academic Press 1976

    Google Scholar 

  10. Laha, R., Rohatgi, V.: Probability theory. New York: Wiley 1979

    Google Scholar 

  11. Loève, M.: Probability theory, vol. I, 4th edition. Berlin Heidelberg New York: Springer 1977

    Google Scholar 

  12. Lukacs, E.: Characteristics functions. London: Griffin 1960

    Google Scholar 

  13. Matheron, G.: Random sets and integral geometry. New York: Wiley 1975

    Google Scholar 

  14. 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)

    Google Scholar 

  15. 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)

    Google Scholar 

  16. 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)

    Google Scholar 

  17. Norberg, T.: Convergence and existence of random set distributions. Ann. Probab. 12, 726–732 (1984)

    Google Scholar 

  18. Norberg, T.: Random capacities and their distributions. Report 1984-03, Dept. of Math., University of Göteborg 1984

  19. Roberts, A., Varberg, D.: Convex functions. New York: Academic Press 1973

    Google Scholar 

  20. Wolfe, S.: On a continuous analogue of the stochastic difference equation 33-1 Stoch. Proc. Appl. 12, 301–312 (1982)

    Google Scholar 

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Authors and Affiliations

  1. Mathematisch Instituut, Katholieke Universiteit, Toernooiveld 5, 6525 ED, Nijmegen, The Netherlands

    Gerard Gerritse

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  1. Gerard Gerritse
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Additional information

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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  • Received: 01 November 1983

  • Revised: 12 December 1985

  • Issue Date: April 1986

  • DOI: https://doi.org/10.1007/BF00343894

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Keywords

  • Stochastic Process
  • Probability Theory
  • Statistical Theory
  • Random Vector
  • Poisson Process
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