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Extremes of vector-valued Gaussian processes: Exact asymptotics

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  • Dȩbicki, Krzysztof
  • Hashorva, Enkelejd
  • Ji, Lanpeng
  • Tabiś, Kamil

Abstract

Let {Xi(t),t≥0},1≤i≤n be mutually independent centered Gaussian processes with almost surely continuous sample paths. We derive the exact asymptotics of P(∃t∈[0,T]∀i=1,…,nXi(t)>u) as u→∞, for both locally stationary Xi’s and Xi’s with a non-constant generalized variance function. Additionally, we analyze properties of multidimensional counterparts of the Pickands and Piterbarg constants that appear in the derived asymptotics. Important by-products of this contribution are the vector-process extensions of the Piterbarg inequality, the Borell–TIS inequality, the Slepian lemma and the Pickands–Piterbarg lemma which are the main pillars of the extremal theory of vector-valued Gaussian processes.

Suggested Citation

  • Dȩbicki, Krzysztof & Hashorva, Enkelejd & Ji, Lanpeng & Tabiś, Kamil, 2015. "Extremes of vector-valued Gaussian processes: Exact asymptotics," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4039-4065.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:11:p:4039-4065
    DOI: 10.1016/j.spa.2015.05.015
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    References listed on IDEAS

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    Cited by:

    1. Bai, Long, 2020. "Extremes of standard multifractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 159(C).
    2. Pingjin Deng, 2018. "The Joint Distribution of Running Maximum of a Slepian Process," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1123-1135, December.
    3. Tang, Linjun & Zheng, Shengchao & Tan, Zhongquan, 2021. "Limit theorem on the pointwise maxima of minimum of vector-valued Gaussian processes," Statistics & Probability Letters, Elsevier, vol. 176(C).
    4. Remco Hofstad & Harsha Honnappa, 2019. "Large deviations of bivariate Gaussian extrema," Queueing Systems: Theory and Applications, Springer, vol. 93(3), pages 333-349, December.
    5. 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.
    6. Dȩbicki, Krzysztof & Hashorva, Enkelejd & Wang, Longmin, 2020. "Extremes of vector-valued Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5802-5837.
    7. Bisewski, Krzysztof & Dȩbicki, Krzysztof & Kriukov, Nikolai, 2023. "Simultaneous ruin probability for multivariate Gaussian risk model," Stochastic Processes and their Applications, Elsevier, vol. 160(C), pages 386-408.
    8. Dȩbicki, Krzysztof & Hashorva, Enkelejd & Ji, Lanpeng & Rolski, Tomasz, 2018. "Extremal behavior of hitting a cone by correlated Brownian motion with drift," Stochastic Processes and their Applications, Elsevier, vol. 128(12), pages 4171-4206.
    9. K. Dębicki & K. M. Kosiński, 2018. "An Erdös–Révész Type Law of the Iterated Logarithm for Order Statistics of a Stationary Gaussian Process," Journal of Theoretical Probability, Springer, vol. 31(1), pages 579-597, March.
    10. Schol, Dennis & Vlasiou, Maria & Zwart, Bert, 2023. "Tail asymptotics for the delay in a Brownian fork-join queue," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 99-138.

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