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Extremes of Gaussian Processes with Maximal Variance near the Boundary Points

Author

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  • Enkelejd Hashorva

    (University of Bern)

  • Jürg Hüsler

    (University of Bern)

Abstract

Let X(t), t∈[0,1], be a Gaussian process with continuous paths with mean zero and nonconstant variance. The largest values of the Gaussian process occur in the neighborhood of the points of maximum variance. If there is a unique fixed point t0 in the interval [0,1], the behavior of P{supt∈[0,1] X(t)>u} is known for u→∞. We investigate the case where the unique point t0 = tu depends on u and tends to the boundary. This is reasonable for a family of Gaussian processes Xu(t) depending on u, which have for each u such a unique point tu tending to the boundary as u→∞. We derive the asymptotic behavior of P{supt∈[0,1] X(t)>u}, depending on the rate as tu tends to 0 or 1. Some applications are mentioned and the computation of a particular case is used to compare simulated probabilities with the asymptotic formula. We consider the exceedances of such a nonconstant boundary by a Ornstein-Uhlenbeck process. It shows the difficulties to simulate such rare events, when u is large.

Suggested Citation

  • Enkelejd Hashorva & Jürg Hüsler, 2000. "Extremes of Gaussian Processes with Maximal Variance near the Boundary Points," Methodology and Computing in Applied Probability, Springer, vol. 2(3), pages 255-269, September.
    • Handle: RePEc:spr:metcap:v:2:y:2000:i:3:d:10.1023_a:1010029228490
    DOI: 10.1023/A:1010029228490
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    References listed on IDEAS

    as
    1. Hüsler, J. & Piterbarg, V., 1999. "Extremes of a certain class of Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 83(2), pages 257-271, October.
    2. Wolfgang Bischoff & Frank Miller, 2000. "Asymptotically Optimal Tests and Optimal Designs for Testing the Mean in Regression Models with Applications to Change-Point Problems," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(4), pages 658-679, December.
    3. Hüsler, J., 1999. "Extremes of Gaussian processes, on results of Piterbarg and Seleznjev," Statistics & Probability Letters, Elsevier, vol. 44(3), pages 251-258, September.
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    Cited by:

    1. Zhongquan Tan & Enkelejd Hashorva, 2014. "On Piterbarg Max-Discretisation Theorem for Standardised Maximum of Stationary Gaussian Processes," Methodology and Computing in Applied Probability, Springer, vol. 16(1), pages 169-185, March.
    2. Krzysztof Dȩbicki & Zbigniew Michna & Xiaofan Peng, 2019. "Approximation of Sojourn Times of Gaussian Processes," Methodology and Computing in Applied Probability, Springer, vol. 21(4), pages 1183-1213, December.
    3. Long Bai & Krzysztof Dȩbicki & Enkelejd Hashorva & Li Luo, 2018. "On Generalised Piterbarg Constants," Methodology and Computing in Applied Probability, Springer, vol. 20(1), pages 137-164, March.

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