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On the extreme order statistics for a stationary sequence

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  • Hsing, Tailen
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    Abstract

    Suppose that {[xi]j} is a strictly stationary sequence which satisfies the strong mixing condition. Denote by M(k)n the kth largest value of [xi]1,[xi]2,...,[xi]n, and {[upsilon]n(·)} a sequence of normalizing functions for which P[M(1)n[less-than-or-equals, slant][upsilon]n(x)]converges weakly to a continuous distribution G(x). It is shown that if for some k=2,3,...,P[M(k)n[less-than-or-equals, slant][upsilon]n(x)] converges for each x, then there exist probabilities p1,...,pk-1 such that P[M(j)n[less-than-or-equals, slant][upsilon]n(x)] converges weakly to for j=2,...,k, where natural interpretations can be given for the pj. This generalizes certain results due to Dziubdziela (1984) and Hsing, Hüsler and Leadbetter (1986). It is further demonstrated that, with minor modification, the technique can be extended to study the joint limiting distribution of the order statistics. In particular, Theorem 1 of Welsch (1972) is generalized, and some links between the convergence of the order statistics and that of certain point processes are established.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 29 (1988)
    Issue (Month): 1 ()
    Pages: 155-169

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    Handle: RePEc:eee:spapps:v:29:y:1988:i:1:p:155-169

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    Related research

    Keywords: extreme values point processes weak convergence;

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    Cited by:
    1. Hashorva, Enkelejd, 2007. "On the asymptotic distribution of certain bivariate reinsurance treaties," Insurance: Mathematics and Economics, Elsevier, vol. 40(2), pages 200-208, March.
    2. Novak, S. Y., 2002. "Multilevel clustering of extremes," Stochastic Processes and their Applications, Elsevier, vol. 97(1), pages 59-75, January.
    3. Hashorva, Enkelejd, 2003. "On the number of near-maximum insurance claim under dependence," Insurance: Mathematics and Economics, Elsevier, vol. 32(1), pages 37-49, February.
    4. Davis, Richard A. & Mikosch, Thomas & Zhao, Yuwei, 2013. "Measures of serial extremal dependence and their estimation," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2575-2602.

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