Journal of Theoretical Probability

, Volume 31, Issue 2, pp 853–866 | Cite as

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

  • Luísa Pereira


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 \(\left\{ X_n\right\} _{n\ge 1}\) 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 \(\left\{ X_{n}\right\} _{n\ge 1}\), 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 \(\left\{ X_{n}\right\} _{n\ge 1}\) with a predetermined probability.


Exceedances Locations of extremes Dependence conditions Point processes 

Mathematics Subject Classification (2010)

60G70 60G55 



The author would like to thank the referees for several corrections and important suggestions which significantly improved this paper. This work was partially supported by National Foundation of Science and Technology through UID/MAT/00212/2013.


  1. 1.
    Balakrishnan, N., Cohen, A.: Order Statistics and Inference. Academic Press Inc, London (1991)MATHGoogle Scholar
  2. 2.
    Davis, R.: Maxima and minima of stationary sequences. Ann. Probab. 7, 453–460 (1979)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Davis, R.: Limit laws for the maximum and minimum of stationary sequences. Z. Wharscheinlichkeits Theorie verw. Gebiete 61, 31–42 (1982)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Davis, R.: Limit laws for upper and lower extremes from stationary mixing sequences. J. Multivar. Anal. 13, 273–286 (1983)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Davis, R.: On upper and lower extremes in stationary sequences. In: Tiago Oliveira, J. (ed.) Statistical Extremes and Applications, NATO ASI Series, vol. 131, pp. 443–460. Springer, Netherlands (1984)Google Scholar
  6. 6.
    Ferreira, H., Scotto, M.: On the asymptotic locations of high values of a stationary sequence. Stat. Probab. Lett. 60, 475–482 (2002)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Hsing, T., Husler, J., Leadbetter, M.R.: On the exceedance point process for a stationary sequence. Probab. Theory Relat. Fields 78, 97–112 (1988)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hsing, T.: On the extreme order statistics for a stationary sequence. Stoch. Process. Appl. 29, 155–169 (1988)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Jakubowski, A.: Asymptotic (\(\text{ r }-1\))-dependent representation for \(r\)th order statistic from a stationary sequence. Stoch. Process. Appl. 46, 29–46 (1993)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kallenberg, O.: Random Measures. Akademische Verlag, Berlin (1975)MATHGoogle Scholar
  11. 11.
    Leadbetter, M.R.: On extreme values in stationary sequences. Wahrscheinlichkeitstheorie and Verwandte Gebiete 2(8), 289–303 (1974)CrossRefMATHGoogle Scholar
  12. 12.
    Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Processes. Springer, Berlin (1983)CrossRefMATHGoogle Scholar
  13. 13.
    Nandagopalan, S.: Multivariate extremes and estimation of the extremal index. Ph.D. Thesis. University of North Carolina at Chapel Hill (1990)Google Scholar
  14. 14.
    Pereira, L., Ferreira, H.: The asymptotic locations of the maximum and minimum of stationary sequences. J. Stat. Plan. Inference 104, 287–295 (2002)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Pereira, L.: The asymptotic location of the maximum of a stationary random field. Stat. Probab. Lett. 79, 2166–2169 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Pereira, L.: Asymptotic location of largest values of a stationary random field. Commun. Stat. Theory Methods 42, 4513–4524 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Pereira, L.: On the maximum and minimum of a stationary random field. In: Lita da Silva, J., Caeiro, F., Natário, I., Braumann C. (eds.) Advances in Regression, Survival Analysis, Extreme Values, Markov Processes and Other Statistical Applications. Studies in Theoretical and Applied Statistics, pp. 337–345. Springer, Berlin, Heidelberg (2013)Google Scholar
  18. 18.
    Pickands, J.: Statistical inference using extreme order statistics. Ann. Stat. 3, 119–131 (1975)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Centro de Matemática e Aplicações (CMA-UBI)Universidade da Beira InteriorCovilhãPortugal

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