On Berman functions

TL;DR

This paper studies Berman functions associated with spectral random fields of stationary max-stable processes, providing new representations and approximation methods useful for Monte Carlo simulations.

Contribution

It introduces novel properties and representations of Berman functions for spectral random fields, enhancing their computational and theoretical understanding.

Findings

Berman functions can be approximated by discrete counterparts

New representations of Berman functions facilitate Monte Carlo simulations

Properties of Berman functions are derived for spectral random fields

Abstract

For fractional Brownian motion with Hurst parameter H the Berman constant is defined. 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, which are presented in this article.