Methodology and Computing in Applied Probability ( IF 0.9 ) Pub Date : 2024-01-05 , DOI: 10.1007/s11009-023-10059-6 Krzysztof Dȩbicki , Enkelejd Hashorva , Zbigniew Michna
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Let \(Z(t)= \exp \left( \sqrt{ 2} B_H(t)- \left|t \right|^{2H}\right) , t\in \mathbb {R}\) with \(B_H(t),t\in \mathbb {R}\) a standard fractional Brownian motion (fBm) with Hurst parameter \(H \in (0,1]\) and define for x non-negative the Berman function
$$\begin{aligned} \mathcal {B}_{Z}(x)= \mathbb {E} \left\{ \frac{ \mathbb {I} \{ \epsilon _0(RZ) > x\}}{ \epsilon _0(RZ)}\right\} \in (0,\infty ), \end{aligned}$$where the random variable R independent of Z has survival function \(1/x,x\geqslant 1\) and
$$\begin{aligned} \epsilon _0(RZ) = \int _{\mathbb {R}} \mathbb {I}{\left\{ RZ(t)> 1\right\} }{dt} . \end{aligned}$$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.
中文翻译：
关于伯曼函数
令\(Z(t)= \exp \left( \sqrt{ 2} B_H(t)- \left|t \right|^{2H}\right) , t\in \mathbb {R}\)与\ (B_H(t),t\in \mathbb {R}\)具有 Hurst 参数\(H \in (0,1]\)的标准分数布朗运动 (fBm) ，并为x定义非负 Berman 函数
$$\begin{对齐} \mathcal {B}_{Z}(x)= \mathbb {E} \left\{ \frac{ \mathbb {I} \{ \epsilon _0(RZ) > x\}} { \epsilon _0(RZ)}\right\} \in (0,\infty ), \end{对齐}$$其中独立于Z的随机变量R具有生存函数\(1/x,x\geqslant 1\)且
$$\begin{对齐} \epsilon _0(RZ) = \int _{\mathbb {R}} \mathbb {I}{\left\{ RZ(t)> 1\right\} }{dt} 。\end{对齐}$$在本文中，我们考虑一般随机场 (rf) Z，它是某些平稳最大稳定 rf X的谱 rf ，并推导相应伯曼函数的属性。特别是，我们表明伯曼函数可以通过相应的离散函数来近似，并导出这些函数的有趣表示，这些函数对于本文中介绍的蒙特卡罗模拟感兴趣。
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