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

Consider a Gaussian random field with a finite Karhunen--Loève expansion of the form Z(u) = ∑ᵢ₌₁ⁿ uᵢ zᵢ, where zᵢ, i=1,…,n, are independent standard normal variables and u=(u₁,…,uₙ)' ranges over an index set M, which is a subset of the unit sphere Sⁿ⁻¹ in Rⁿ. 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.