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On the Asymptotic Locations of the Largest and Smallest Extremes of a Stationary Sequence

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  • Luísa Pereira

    (Universidade da Beira Interior)

Abstract

This paper deals with the asymptotic independence of the normalized kth upper- and rth lower-order statistics and their locations, defined on some strictly stationary sequences $$\left\{ X_n\right\} _{n\ge 1}$$ X n n ≥ 1 admitting clusters of both high and low values. The main result is the asymptotic independence of the joint locations of the k-largest extremes and the joint locations of the r-smallest extremes of $$\left\{ X_{n}\right\} _{n\ge 1}$$ X n n ≥ 1 , which allows us to censor a sample, by ensuring that the set of observations that we selected contains the k-largest and r-smallest order statistics of the stationary sequence $$\left\{ X_{n}\right\} _{n\ge 1}$$ X n n ≥ 1 with a predetermined probability.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:31:y:2018:i:2:d:10.1007_s10959-017-0742-8
    DOI: 10.1007/s10959-017-0742-8
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    References listed on IDEAS

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    1. Ferreira, H. & Scotto, M., 2002. "On the asymptotic location of high values of a stationary sequence," Statistics & Probability Letters, Elsevier, vol. 60(4), pages 475-482, December.
    2. Pereira, L., 2009. "The asymptotic location of the maximum of a stationary random field," Statistics & Probability Letters, Elsevier, vol. 79(20), pages 2166-2169, October.
    3. Hsing, Tailen, 1988. "On the extreme order statistics for a stationary sequence," Stochastic Processes and their Applications, Elsevier, vol. 29(1), pages 155-169.
    4. Jakubowski, Adam, 1993. "Asymptotic (r- 1)-dependent representation for rth order statistic from a stationary sequence," Stochastic Processes and their Applications, Elsevier, vol. 46(1), pages 29-46, May.
    5. Davis, Richard A., 1983. "Limit laws for upper and lower extremes from stationary mixing sequences," Journal of Multivariate Analysis, Elsevier, vol. 13(2), pages 273-286, June.
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    Cited by:

    1. Liu, Huiyan & Tan, Zhongquan, 2022. "Point processes of exceedances by Gaussian random fields with applications to asymptotic locations of extreme order statistics," Statistics & Probability Letters, Elsevier, vol. 189(C).

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