On Berman Functions

Krzysztof Dȩbicki (), Enkelejd Hashorva () and Zbigniew Michna ()
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Krzysztof Dȩbicki: University of Wrocław
Enkelejd Hashorva: University of Lausanne
Zbigniew Michna: Wrocław University of Science and Technology

Methodology and Computing in Applied Probability, 2024, vol. 26, issue 1, 1-27

Abstract: Abstract Let $$Z(t)= \exp \left( \sqrt{ 2} B_H(t)- \left|t \right|^{2H}\right) , t\in \mathbb {R}$$ Z ( t ) = exp 2 B H ( t ) - t 2 H , t ∈ R with $$B_H(t),t\in \mathbb {R}$$ B H ( t ) , t ∈ R a standard fractional Brownian motion (fBm) with Hurst parameter $$H \in (0,1]$$ H ∈ ( 0 , 1 ] and define for x non-negative the Berman function $$\begin{aligned} \mathcal {B}_{Z}(x)= \mathbb {E} \left\{ \frac{ \mathbb {I} \{ \epsilon _0(RZ) > x\}}{ \epsilon _0(RZ)}\right\} \in (0,\infty ), \end{aligned}$$ B Z ( x ) = E I { ϵ 0 ( R Z ) > x } ϵ 0 ( R Z ) ∈ ( 0 , ∞ ) , where the random variable R independent of Z has survival function $$1/x,x\geqslant 1$$ 1 / x , x ⩾ 1 and $$\begin{aligned} \epsilon _0(RZ) = \int _{\mathbb {R}} \mathbb {I}{\left\{ RZ(t)> 1\right\} }{dt} . \end{aligned}$$ ϵ 0 ( R Z ) = ∫ R I R Z ( t ) > 1 dt . In this paper we consider a general random field (rf) Z that is a spectral rf of some stationary max-stable rf X and derive the properties of the corresponding Berman functions. In particular, we show that Berman functions can be approximated by the corresponding discrete ones and derive interesting representations of those functions which are of interest for Monte Carlo simulations presented in this article.

Keywords: Berman functions; Pickands constants; Max-stable random fields; Simulations; Primary 60G15; Secondary 60G70 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s11009-023-10059-6

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