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On the continuity of Pickands constants

Part of: Stochastic processes

Published online by Cambridge University Press:  18 January 2022

Krzysztof Dębicki ,
Enkelejd Hashorva  and
Zbigniew Michna
Krzysztof Dębicki*
Affiliation:
University of Wrocław
Enkelejd Hashorva*
Affiliation:
University of Lausanne
Zbigniew Michna*
Affiliation:
Wrocław University of Science and Technology
*
*Postal address: Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. Email address: Krzysztof.Debicki@math.uni.wroc.pl
**Postal address: Department of Actuarial Science, University of Lausanne, UNIL-Dorigny, 1015 Lausanne, Switzerland. Email address: Enkelejd.Hashorva@unil.ch
***Postal address: Department of Operations Research and Business Intelligence, Wrocław University of Science and Technology, wybrzeze Stanislawa Wyspianskiego 27, 50-370 Wrocław, Poland. Article partially completed at Wrocław University of Economics and Business. Email address: zbigniew.michna@pwr.edu.pl
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Abstract

For a non-negative separable random field Z(t), $t\in \mathbb{R}^d$ , satisfying some mild assumptions, we show that $ H_Z^\delta =\lim_{{T} \to \infty} ({1}/{T^d}) \mathbb{E}\{{\sup_{ t\in [0,T]^d \cap \delta \mathbb{Z}^d } Z(t) }\} <\infty$ for $\delta \ge 0$ , where $0 \mathbb{Z}^d\,:\!=\,\mathbb{R}^d$ , and prove that $H_Z^0$ can be approximated by $H_Z^\delta$ if $\delta$ tends to 0. These results extend the classical findings for Pickands constants $H_{Z}^\delta$ , defined for $Z(t)= \exp( \sqrt{ 2} B_\alpha (t)- \lvert {t} \rvert^{2\alpha })$ , $t\in \mathbb{R}$ , with $B_\alpha$ a standard fractional Brownian motion with Hurst parameter $\alpha \in (0,1]$ . The continuity of $H_{Z}^\delta$ at $\delta=0$ is additionally shown for two particular extensions of Pickands constants.

Keywords

Pickands constantsdiscrete approximationlocally stationary Gaussian random fieldsmax-stable random fieldsextremal index

MSC classification

Primary: 60G10: Stationary processes
Secondary: 60G70: Extreme value theory; extremal processes
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 1 , March 2022 , pp. 187 - 201
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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