On Berman Functions

Abstract

Let Z(t)=exp2BH(t)-t2H,t∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Z(t)= \\exp \\left( \\sqrt{ 2} B_H(t)- \\left|t \\right|^{2H}\\right) , t\\in \\mathbb {R}$$\\end{document} with BH(t),t∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_H(t),t\\in \\mathbb {R}$$\\end{document} a standard fractional Brownian motion (fBm) with Hurst parameter H∈(0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H \\in (0,1]$$\\end{document} and define for x non-negative the Berman function BZ(x)=EI{ϵ0(RZ)>x}ϵ0(RZ)∈(0,∞),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mathcal {B}_{Z}(x)= \\mathbb {E} \\left\\{ \\frac{ \\mathbb {I} \\{ \\epsilon _0(RZ) > x\\}}{ \\epsilon _0(RZ)}\\right\\} \\in (0,\\infty ), \\end{aligned}$$\\end{document}where the random variable R independent of Z has survival function 1/x,x⩾1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/x,x\\geqslant 1$$\\end{document} and ϵ0(RZ)=∫RIRZ(t)>1dt.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\epsilon _0(RZ) = \\int _{\\mathbb {R}} \\mathbb {I}{\\left\\{ RZ(t)> 1\\right\\} }{dt} . \\end{aligned}$$\\end{document}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.