On the equivalence of the tube and Euler characteristic methods for the distribution of the maximum of Gaussian fields over piecewise smooth domains

Abstract

Consider a Gaussian random field with a finite Karhunen--Loève expansion of the form $Z(u) = \sum_{i=1}^n u_i z_i$, where $z_i$, $i=1,\ldots,n,$ are independent standard normal variables and $u=(u_1,\ldots,u_n)'$ ranges over an index set $M$, which is a subset of the unit sphere $S^{n-1}$ in $R^n$. Under a very general assumption that $M$ is a manifold with a piecewise smooth boundary, we prove the validity and the equivalence of two currently available methods for obtaining the asymptotic expansion of the tail probability of the maximum of $Z(u)$. One is the tube method, where the volume of the tube around the index set $M$ is evaluated. The other is the Euler characteristic method, where the expectation for the Euler characteristic of the excursion set is evaluated. General discussion on this equivalence was given in a recent paper by R. J. Adler. In order to show the equivalence we prove a version of the Morse theorem for a manifold with a piecewise smooth boundary.