Asymptotic formula for the tail of the maximum of smooth Stationary Gaussian fields on non locally convex sets

TL;DR

This paper extends the asymptotic analysis of the maximum distribution of smooth stationary Gaussian fields to non-locally convex sets, providing full expansions in two dimensions and generalizations in higher dimensions.

Contribution

It generalizes existing results from locally convex to non-locally convex sets by establishing a Steiner formula and linking it to the tail distribution of the maximum.

Findings

Full asymptotic expansion in 2D for non-convex sets

Steiner formula relates set geometry to maximum tail behavior

Examples demonstrate applicability in higher dimensions

Abstract

In this paper we consider the distribution of the maximum of a Gaussian field defined on non locally convex sets. Adler and Taylor or Aza\"\i s and Wschebor give the expansions in the locally convex case. The present paper generalizes their results to the non locally convex case by giving a full expansion in dimension 2 and some generalizations in higher dimension. For a given class of sets, a Steiner formula is established and the correspondence between this formula and the tail of the maximum is proved. The main tool is a recent result of Aza\"\i s and Wschebor that shows that under some conditions the excursion set is close to a ball with a random radius. Examples are given in dimension 2 and higher.