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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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    1. Nourdin, Ivan & Peccati, Giovanni & Viens, Frederi G., 2014. "Comparison inequalities on Wiener space," Stochastic Processes and their Applications, Elsevier, vol. 124(4), pages 1566-1581.
    2. Dieker, A.B., 2005. "Extremes of Gaussian processes over an infinite horizon," Stochastic Processes and their Applications, Elsevier, vol. 115(2), pages 207-248, February.
    3. 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.
    4. Dëbicki, Krzysztof & Kisowski, Pawel, 2008. "A note on upper estimates for Pickands constants," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2046-2051, October.
    5. Debicki, K. & Kosinski, K.M. & Mandjes, M. & Rolski, T., 2010. "Extremes of multidimensional Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2289-2301, December.
    6. Worsley, K. J. & Friston, K. J., 2000. "A test for a conjunction," Statistics & Probability Letters, Elsevier, vol. 47(2), pages 135-140, April.
    7. Krzysztof Dȩbicki & Enkelejd Hashorva & Lanpeng Ji & Chengxiu Ling, 2015. "Extremes of order statistics of stationary processes," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(2), pages 229-248, June.
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    Cited by:

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    3. Bai, Long, 2020. "Extremes of standard multifractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 159(C).
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    6. 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.
    7. 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.
    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.
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