On Berman Functions

Abstract

Let $Z\left(t\right)=exp\left(\sqrt{2},B_H,\left(t\right),-,{\left(t\right)}^{2H}\right),t\in R$ with $B_H\left(t\right),t\in R$ a standard fractional Brownian motion (fBm) with Hurst parameter $H\in (0,1]$ and define for x non-negative the Berman function $\begin{array}{c}{B}_Z\left(x\right)=E\left(\frac{I\{{ϵ}_0\left(RZ\right)>x\}}{{ϵ}_0\left(RZ\right)}\right)\in (0,\infty ),\end{array}$where the random variable R independent of Z has survival function $1/x,x⩾1$ and $\begin{array}{c}{ϵ}_0\left(RZ\right)={\int }_{R}I\left(R,Z,\left(,t,\right),>,1\right)\mathrm{dt}.\end{array}$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; Max-stable random fields; Pickands constants; Simulations.