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On the extreme order statistics for a stationary sequence

Author

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  • Hsing, Tailen
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.

Suggested Citation

  • Hsing, Tailen, 1988. "On the extreme order statistics for a stationary sequence," Stochastic Processes and their Applications, Elsevier, vol. 29(1), pages 155-169.
    • Handle: RePEc:eee:spapps:v:29:y:1988:i:1:p:155-169
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    Citations

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    Cited by:

    1. Soja-Kukieła, Natalia, 2017. "Asymptotics of the order statistics for a process with a regenerative structure," Statistics & Probability Letters, Elsevier, vol. 131(C), pages 108-115.
    2. Novak, S. Y., 2002. "Multilevel clustering of extremes," Stochastic Processes and their Applications, Elsevier, vol. 97(1), pages 59-75, January.
    3. 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.
    4. Hugo C. Winter & Jonathan A. Tawn, 2016. "Modelling heatwaves in central France: a case-study in extremal dependence," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 65(3), pages 345-365, April.
    5. Ji, Lanpeng & Peng, Xiaofan, 2023. "Extreme value theory for a sequence of suprema of a class of Gaussian processes with trend," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 418-452.
    6. Luísa Pereira, 2018. "On the Asymptotic Locations of the Largest and Smallest Extremes of a Stationary Sequence," Journal of Theoretical Probability, Springer, vol. 31(2), pages 853-866, June.
    7. Hashorva, Enkelejd, 2007. "On the asymptotic distribution of certain bivariate reinsurance treaties," Insurance: Mathematics and Economics, Elsevier, vol. 40(2), pages 200-208, March.
    8. Matthew J. Schneider & Rufus Rankin & Prabir Burman & Alexander Aue, 2024. "Benchmarking M6 Competitors: An Analysis of Financial Metrics and Discussion of Incentives," Papers 2406.19105, arXiv.org, revised Aug 2024.
    9. 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.
    10. Chenavier, Nicolas, 2014. "A general study of extremes of stationary tessellations with examples," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 2917-2953.

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