IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v29y1988i1p155-169.html
   My bibliography  Save this article

On the extreme order statistics for a stationary sequence

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/0304-4149(88)90035-X
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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. 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.
    8. 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.
    9. 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.
    10. Hashorva, Enkelejd, 2007. "On the asymptotic distribution of certain bivariate reinsurance treaties," Insurance: Mathematics and Economics, Elsevier, vol. 40(2), pages 200-208, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:29:y:1988:i:1:p:155-169. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.