The tail of the maximum of smooth Gaussian fields on fractal sets

TL;DR

This paper investigates the asymptotic behavior of the tail distribution of the maximum of smooth Gaussian fields on fractal sets, using Rice formula to analyze local maxima.

Contribution

It provides the first detailed asymptotic analysis of maximum distributions of Gaussian fields on fractal sets, extending classical results to complex geometries.

Findings

Asymptotic tail behavior of maximum distribution characterized

Rice formula effectively used for moments of local maxima

Results applicable to fractal geometries in Gaussian fields

Abstract

We study the probability distribution of the maximum $M_S $ of a smooth stationary Gaussian field defined on a fractal subset $S$ of $\R^n$. Our main result is the equivalent of the asymptotic behavior of the tail of the distribution $\P(M_S>u)$ as $u\rightarrow +\infty.$ The basic tool is Rice formula for the moments of the number of local maxima of a random field.