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Stationary Self-Similar Extremal Processes and Random Semicontinuous Functions

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Dependence in Probability and Statistics

Part of the book series: Progress in Probability and Statistics ((PRPR,volume 11))

Abstract

The main topic of this survey is the characterization of all limiting processes of

, where (ξ k ) ∞ k = 1 may be any stationary sequence of random variables. The limits are identified as the stationary self-similar extremal processes, and some of their properties are investigated.

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References

  1. Adler, R.J. (1978): Weak convergence results for extremal processes generated by dependent random variables. Ann. Probab. 6 660–667.

    Article  MATH  MathSciNet  Google Scholar 

  2. Balkema, A.A. & S.I. Resnick (1977): Max-infinite divisibility. Z. Wahrsch. verw. Gebiete 14 309–319.

    MATH  MathSciNet  Google Scholar 

  3. Berman, S.M. (1964): Limit theorems for the maximum term in stationary sequences. Ann. Math. Statist. 35 502–516.

    Article  MATH  MathSciNet  Google Scholar 

  4. DE Haan, L. (1970): On regular variation and its application to the weak convergence of sample extremes. Math. Centre Tracts 32, Mathematisch Centrum, Amsterdam.

    Google Scholar 

  5. DE Haan, L. & S.I. Resntck (1977): Limit theory for multivariate sample extremes. Z. Wahrsch. verw. Gebiete 40 317–337.

    Article  MATH  Google Scholar 

  6. Deheuvels, P. (1978): Caractérisation complète des lois extremes multivariées et de la convergence des types extremes. Pub. Inst. Stat. Univ. Paris 23.3–4 1–36.

    Google Scholar 

  7. Deheuvels, P. (1981): The strong approximation of extremal processes. Z. Wahrsch. verw. Gebiete 58 1–6.

    Article  MATH  MathSciNet  Google Scholar 

  8. Deheuvels, P. (1983): The strong approximation of extremal processes (Il). Z. Wahrsch. verw. Gebiete 62 7–15.

    Article  MATH  MathSciNet  Google Scholar 

  9. Dwass, M. (1964): Extremal processes. Ann. Math. Statist. 35 1718–1725.

    Google Scholar 

  10. Fisher, R.A. & L.H.C. Tippett (1928): Limiting forms of the frequency distribution of the largest or smallest member of a sample. Proc. Cambridge Philos. Soc. 242 180–190.

    Article  Google Scholar 

  11. Gerritse, G. (1983): Supremum self-decomposable random vectors. Report 8341, Dept of Mathematics, Catholic Univ., Nijmegen. To appear in Z. Wahrsch. verw. Gebiete.

    Google Scholar 

  12. Gerritse, G. (1985): Lattice-valued semicontinuous functions. Report 8532, Dept of Mathematics, Catholic Univ., Nijmegen.

    Google Scholar 

  13. Gierz, G., K.H. Hofmann, K. KEtMel, J.D. Lawson, M. Misi.ovE & D.S. Scorr (1980): A compendium of continuous lattices. Springer, Berlin.

    Book  MATH  Google Scholar 

  14. Gnedenko, B.V. (1943): Sur la distribution limite du terme maximum d’une série aléatoire. Ann. of Math. 44 423–453.

    Article  MATH  MathSciNet  Google Scholar 

  15. Goldie, C. (1983): On records and related topics in probability theory. Thesis, University of Sussex.

    Google Scholar 

  16. Hofmann, K.H. & M.W. Mtstove (1981): Local compactness and continuous lattices. In: Continuous lattices (Proceedings, Bremen 1979), eds. B. Banaschewski & R.-E. Hofmann. Lecture Notes in Math. 871, Springer, Berlin.

    Google Scholar 

  17. Lamperti, J. (1962): Semi-stable stochastic processes. Trans. Amer. Math. Soc. 104 62–78.

    Article  MATH  MathSciNet  Google Scholar 

  18. Lamperti, J. (1964): On extreme order statistics. Ann. Math. Statist. 35 1726–1737.

    Article  MATH  MathSciNet  Google Scholar 

  19. Leadbetter, M.R., G. Lindgren & H. RoorzfN (1983): Extremes and related properties of random sequences and processes. Springer, New York, Heidelberg, Berlin.

    Google Scholar 

  20. Loynes, M.R. (1965): Extreme values in uniformly mixing stationary stochastic processes. Ann. Math. Statist. 36 993–999.

    Article  MATH  MathSciNet  Google Scholar 

  21. Matheron, G. (1975): Random sets and integral geometry. Wiley, New York.

    MATH  Google Scholar 

  22. Mori, T. (1977): Limit distributions of two-dimensional point processes generated by strong mixing sequences. Yokohama Math. J. 25 155–168.

    MATH  MathSciNet  Google Scholar 

  23. Mori, T. & H. Oodaira (1976): A functional law of the iterated logarithm for sample sequences. Yokohama Math. J. 24 35–49.

    MATH  MathSciNet  Google Scholar 

  24. Newell, G.F. (1964): Asymptotic extremes for m-dependent random variables. Ann. Math. Statist. 35 1322–1325.

    Article  MATH  MathSciNet  Google Scholar 

  25. Norberg, T. (1984): Random capacities and their distributions. Preprint 1984–03, Chalmers University of Technology, Göteborg. To appear in Z. Wahrsch. verw, Gebiete.

    Google Scholar 

  26. O’Brien, G.L. (1984): Extreme values for stationary and Markov sequences. Preprint, York University.

    Google Scholar 

  27. Obrien, G.L., P.J.J.F. Torfs & W. Vervaat (1985): Stationary self-similar extremal processes. Forthcoming.

    Google Scholar 

  28. O’Brien, G.L. & W. Vervaat (1985): Self-similar processes with stationary increments generated by point processes. Ann. Probab. 13 28–52.

    Google Scholar 

  29. Penot, J.P. & M. Thera (1982): Semi-continuous mappings in general topology. Arch. Math. 38 158–166.

    Article  MATH  MathSciNet  Google Scholar 

  30. Pickands Iii, J. (1971): The two-dimensional Poisson process and extremal processes. J. App!. Probab. 8 745–756.

    Article  MATH  MathSciNet  Google Scholar 

  31. Resnick, S.I. (1975): Weak convergence to extremal processes. Ann. Probab. 3 951–960.

    Article  MATH  MathSciNet  Google Scholar 

  32. Resnick, S.I. (1985): Point processes, regular variation and weak convergence. Adv. Appl. Probab. (to appear).

    Google Scholar 

  33. Salinetti, G. & R.J.-B. Wets (1985): On the convergence in distribution of measurable multifunctions, normal integrands, stochastic processes and stochastic infima. Math, Oper. Res. (to appear).

    Google Scholar 

  34. Salinetti, G., W. Vervaat & R.J.-B. Wets (1985): On the convergence in probability of random sets (measurable multifunctions). Math. Oper. Res. (to appear).

    Google Scholar 

  35. Vervaat, W. (1981): The structure of limit theorems in probability. Lecture notes, Catholic University, Leuven & Catholic University, Nijmegen.

    Google Scholar 

  36. Vervaat, W. (1982): Random semicontinuous functions and extremal processes. Handwritten manuscript. Being revised.

    Google Scholar 

  37. Vervaat, W. (1985): Sample path properties of self-similar processes with stationary increments. Ann. Probab. 13 1–27.

    Article  MATH  MathSciNet  Google Scholar 

  38. Weissman, I. (1975a): On location and scale functions of a class of limiting processes with application to extreme value theory. Ann. Probab. 3 178–1R1.

    Google Scholar 

  39. Weissman, I. (19756): Multivariate extremal processes generated by independent non-identically distributed random variables. J. App!. Probab. 12 477–487.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Catholic University, Toernooiveld 5, 6525 ED, Nijmegen, The Netherlands

    Wim Vervaat

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  1. Wim Vervaat
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Editor information

Editors and Affiliations

  1. Institut für Mathematische Stochastik, Universität Freiburg, 7800, Freiburg i. Br., Federal Republic of Germany

    Ernst Eberlein

  2. Department of Mathematics, Boston University, 02215, Boston, MA, USA

    Murad S. Taqqu

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© 1986 Springer Science+Business Media New York

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Vervaat, W. (1986). Stationary Self-Similar Extremal Processes and Random Semicontinuous Functions. In: Eberlein, E., Taqqu, M.S. (eds) Dependence in Probability and Statistics. Progress in Probability and Statistics, vol 11. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4615-8162-8_22

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  • DOI: https://doi.org/10.1007/978-1-4615-8162-8_22

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4615-8163-5

  • Online ISBN: 978-1-4615-8162-8

  • eBook Packages: Springer Book Archive

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