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Exact asymptotics of supremum of a stationary Gaussian process over a random interval

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  • Arendarczyk, Marek
  • Dȩbicki, Krzysztof

Abstract

Let {X(t):t∈[0,∞)} be a centered stationary Gaussian process. We study the exact asymptotics of P(sups∈[0,T]X(s)>u), as u→∞, where T is an independent of {X(t)} nonnegative random variable. It appears that the heaviness of T impacts the form of the asymptotics, leading to three scenarios: the case of integrable T, the case of T having regularly varying tail distribution with parameter λ∈(0,1) and the case of T having slowly varying tail distribution.

Suggested Citation

  • Arendarczyk, Marek & Dȩbicki, Krzysztof, 2012. "Exact asymptotics of supremum of a stationary Gaussian process over a random interval," Statistics & Probability Letters, Elsevier, vol. 82(3), pages 645-652.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:3:p:645-652
    DOI: 10.1016/j.spl.2011.11.015
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    References listed on IDEAS

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    1. Debicki, Krzystof & Zwart, Bert & Borst, Sem, 2004. "The supremum of a Gaussian process over a random interval," Statistics & Probability Letters, Elsevier, vol. 68(3), pages 221-234, July.
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    Cited by:

    1. Dȩbicki, Krzysztof & Hashorva, Enkelejd & Ji, Lanpeng & Tabiś, Kamil, 2014. "On the probability of conjunctions of stationary Gaussian processes," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 141-148.
    2. Yang, Yang & Hashorva, Enkelejd, 2013. "Extremes and products of multivariate AC-product risks," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 312-319.
    3. Tan, Zhongquan & Hashorva, Enkelejd, 2013. "Exact asymptotics and limit theorems for supremum of stationary χ-processes over a random interval," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 2983-2998.
    4. Popivoda, Goran & Stamatović, Siniša, 2016. "Extremes of Gaussian fields with a smooth random variance," Statistics & Probability Letters, Elsevier, vol. 110(C), pages 185-190.
    5. Tan, Zhongquan, 2013. "An almost sure limit theorem for the maxima of smooth stationary Gaussian processes," Statistics & Probability Letters, Elsevier, vol. 83(9), pages 2135-2141.

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    1. Tan, Zhongquan & Hashorva, Enkelejd, 2013. "Exact asymptotics and limit theorems for supremum of stationary χ-processes over a random interval," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 2983-2998.

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