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Approximation of Supremum of Max-Stable Stationary Processes & Pickands Constants

  • Krzysztof Dȩbicki
  • Enkelejd HashorvaEmail author
Article
  • 36 Downloads

Abstract

Let \(X(t),t\in \mathbb {R}\) be a stochastically continuous stationary max-stable process with Fréchet marginals \(\Phi _\alpha , \alpha >0\) and set \(M_X(T)=\sup _{t \in [0,T]} X(t),T>0\). In the light of the seminal articles (Samorodnitsky in Ann Probab 32(2):1438–1468, 2004; Adv Appl Probab 36(3):805–823, 2004), it follows that \(A_T=M_X(T)/T^{1/\alpha }\) converges in distribution as \(T\rightarrow \infty \) to \( \mathcal {H}^{1/\alpha } X(1)\), where \( \mathcal {H}\) is the Pickands constant corresponding to the spectral process Z of X. In this contribution, we derive explicit formulas for \( \mathcal {H}\) in terms of Z and show necessary and sufficient conditions for its positivity. From our analysis, it follows that \(A_T^\beta ,T>0\) is uniformly integrable for any \(\beta \in (0,\alpha )\). For Brown–Resnick X, we show the validity of the celebrated Slepian inequality and discuss the finiteness of Piterbarg constants.

Keywords

Max-stable process Spectral tail process Gaussian processes with stationary increments Lévy processes Pickands constants Piterbarg constants Slepian inequality Growth of supremum 

Mathematics Subject Classification (2010)

Primary 60G15 Secondary 60G70 

Notes

Acknowledgements

Many thanks to Parthanil Roy for discussions and suggestion of the key reference [20]. We thank the referees for numerous suggestions that improved the original manuscript. EH was supported by SNSF Grant 200021-175752/1. KD was partially supported by NCN Grant No 2015/17/B/ST1/01102 (2016-2019).

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of WrocławWrocławPoland
  2. 2.Department of Actuarial ScienceUniversity of LausanneLausanneSwitzerland

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