Skip to main content
SpringerLink
Log in

The limit theorems for maxima of stationary Gaussian processes with random Index

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

Let {X(t), t ≥ 0} be a standard (zero-mean, unit-variance) stationary Gaussian process with correlation function r(·) and continuous sample paths. In this paper, we consider the maxima M(T) = max{X(t), ∀t ∈ [0, T]} with random index T T , where T T /T converges to a non-degenerate distribution or to a positive random variable in probability, and show that the limit distribution of M(T T ) exists under some additional conditions related to the correlation function r(·).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barakat, H. M., Nigm, E. M.: Weak limit of sample range and midrange for random sample sizes. Egyptian Statist. J., 34, 45–58 (1990)

    MathSciNet  Google Scholar 

  2. Barakat, H. M., Nigm, E. M.: Convergence of random extremal quotient and product. J. Statist. Plann. Inference, 81, 209–221 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Barakat, H. M., Nigm, E. M.: Extreme order statistics under power normalization and random sample size. Kuwait J. Sci. Eng., 29, 27–41 (2002)

    MATH  MathSciNet  Google Scholar 

  4. Barakat, H. M., El-Shandidy, M. A.: On general asymptotic behaviour of order statistics with random index. Bull. Malays. Math. Sci. Soc., 27, 169–183 (2004)

    MATH  MathSciNet  Google Scholar 

  5. Berman, M. S.: Limit theorems for the maximum term in stationary sequences. Ann. Inst. Statist. Math., 35, 502–516 (1966)

    Article  Google Scholar 

  6. Berman, M. S.: Sojourns and Extremes of Stochastic Processes, Wadsworth & Brooks/Cole, Boston, 1992

    MATH  Google Scholar 

  7. Dębicki, K., Kisowski, P.: A note on upper estimates for Pickands constants. Statist. Probab. Lett., 78, 2046–2051 (2009)

    Article  Google Scholar 

  8. Dorea, C., Goncalves, C.: Asymptotic distribution of extremes of randomly indexed random variables. Extremes, 2, 95–109 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ferreira, H.: Extremes of a random number of variables from periodic sequences. J. Statist. Plann. Inference, 45, 133–141 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  10. Freitas, A., Hüsler, J., Temido, M. G.: Limit laws for maxima of a stationary random sequence with random size. Test, 21, 116–131 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Galambos, J.: The distribution of the maximum of a random number of random variables with applications. J. Appl. Probab., 10, 122–129 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  12. Galambos, J.: Limit laws for mixtures with applications to asymptotic theory of extremes. Z. Wahrscheinlichkeitsth., 32, 197–207 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  13. Galambos, J.: The Asymptotic Theory of Extreme Order Statistics, Wiley, New York, 1978

    MATH  Google Scholar 

  14. Galambos, J.: Random sample sizes: limit theorems and characterizations. In: Probability Theory and Applic-ations (ed. J. Galambos and I. Kftai), Kluwer, Dordrecht, 1992

    Chapter  Google Scholar 

  15. Galambos, J.: About the development of mathematical theory of extremes during the last half-century. Theory Probab. Appl., 39, 272–293 (1994)

    MATH  MathSciNet  Google Scholar 

  16. Hashorva, E., Hüsler, J.: Extremes of Gaussian processes with maximal variance near the boundary points. Methodol. Comput. Appl. Probab., 2, 255–269 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Leadbetter, M. R., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Processes, Springer-Verlag, New York, 1983

    Book  MATH  Google Scholar 

  18. Mittal, Y., Ylvisaker, D.: Limit distribution for the maximum of stationary Gaussian processes. Stochastic Process. Appl., 3, 1–18 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  19. Peng, Z., Jiang, Q., Nadarajah, S.: Limiting distributions of extreme order statistics under power normalization and random index. Stochastics, 84, 553–560 (2012)

    MATH  MathSciNet  Google Scholar 

  20. Piterbarg, V. I.: Asymptotic Methods in the Theory of Gaussian Processes and Fields, AMS, Providence, 1996

    MATH  Google Scholar 

  21. Silvestrov, D. S., Teugels, J. L.: Limit theorems for extremes with random sample size. Adv. in Appl. Probab., 30, 777–806 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  22. Tan, Z., Hashorva, E.: Exact tail asymptotics for the supremum of strongly dependent Gaussian processes over a random interval. Lithuanian Math. J., 53, 91–102 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  23. Tan, Z., Hashorva, E., Peng, Z.: Asymptotics of maxima of strongly dependent Gaussian processes. J. Appl. Probab., 49, 1106–1118 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  24. Tan, Z., Tang, L.: The dependence of extremes values of discrete and continuous time strongly dependent Gaussian processes. Stochastics, 86, 60–69 (2014)

    MathSciNet  Google Scholar 

  25. Tan, Z., Wang, Y.: Extremes values of discrete and continuous time strongly dependent Gaussian processes. Comm. Statist. Theory Methods, 42, 2451–2463 (2013)

    Article  MATH  Google Scholar 

  26. Zhuang, G., Peng, Z., Xia, J.: On the limiting distribution of the maximum of weakly dependent Gaussian sequences with random index. Acta Math. Appl. Sin., 31, 1068–1079 (2008)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, 314001, P. R. China

    Zhong Quan Tan

Authors
  1. Zhong Quan Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhong Quan Tan.

Additional information

Supported by National Science Foundation of China (Grant No. 11326175) and Research Start-up Foundation of Jiaxing University (Grant No. 70512021)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tan, Z.Q. The limit theorems for maxima of stationary Gaussian processes with random Index. Acta. Math. Sin.-English Ser. 30, 1021–1032 (2014). https://doi.org/10.1007/s10114-014-2809-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-014-2809-0

Keywords

MR(2010) Subject Classification

Search

Navigation