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Excursion probability of certain non-centered smooth Gaussian random fields

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  • Cheng, Dan

Abstract

Let X={X(t),t∈T} be a non-centered, unit-variance, smooth Gaussian random field indexed on some parameter space T, and let Au(X,T)={t∈T:X(t)≥u} be the excursion set. It is shown that, as u→∞, the excursion probability P{supt∈TX(t)≥u} can be approximated by the expected Euler characteristic of Au(X,T), denoted by E{χ(Au(X,T))}, such that the error is super-exponentially small. The explicit formulae for E{χ(Au(X,T))} are also derived for two cases: (i) T is a rectangle and X−EX is stationary; (ii) T is an N-dimensional sphere and X−EX is isotropic.

Suggested Citation

  • Cheng, Dan, 2016. "Excursion probability of certain non-centered smooth Gaussian random fields," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 883-905.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:3:p:883-905
    DOI: 10.1016/j.spa.2015.10.003
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    References listed on IDEAS

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    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. Azaïs, Jean-Marc & Wschebor, Mario, 2008. "A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail," Stochastic Processes and their Applications, Elsevier, vol. 118(7), pages 1190-1218, July.
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    Cited by:

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    2. Wang, Yizao, 2018. "Extremes of q-Ornstein–Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, vol. 128(9), pages 2979-3005.
    3. Qiao, Wanli, 2021. "Extremes of locally stationary Gaussian and chi fields on manifolds," Stochastic Processes and their Applications, Elsevier, vol. 133(C), pages 166-192.
    4. Chunsheng Ma, 2017. "Time Varying Isotropic Vector Random Fields on Spheres," Journal of Theoretical Probability, Springer, vol. 30(4), pages 1763-1785, December.
    5. Chunsheng Ma & Anatoliy Malyarenko, 2020. "Time-Varying Isotropic Vector Random Fields on Compact Two-Point Homogeneous Spaces," Journal of Theoretical Probability, Springer, vol. 33(1), pages 319-339, March.
    6. Ji, Lanpeng & Liu, Peng & Robert, Stephan, 2019. "Tail asymptotic behavior of the supremum of a class of chi-square processes," Statistics & Probability Letters, Elsevier, vol. 154(C), pages 1-1.
    7. Chunsheng Ma, 2023. "Vector Random Fields on the Probability Simplex with Metric-Dependent Covariance Matrix Functions," Journal of Theoretical Probability, Springer, vol. 36(3), pages 1922-1938, September.
    8. Lan, Xiaohong & Xiao, Yimin, 2018. "Strong local nondeterminism of spherical fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 135(C), pages 44-50.

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