"Maximum of Gaussian Field on Piecewise Smooth Domain: Equivalence of Tube Method and Euler Characteristic Method"

Consider a Gaussian random field Z(u) with mean 0, variance 1, and finite Karhunen-Loeve expansion. Under a very general assumption that the index set M is a manifold with piecewise smooth boundary, we prove the validity and the equivalence of two currently available methods for obtaining the asymptotic expansion of tail probability of the maximum of Z(u). One is the tube method, where the volume of tube around the index set M is evaluated. The other is the Euler characteristic method, where the expectation for the Euler characteristic of excursion set is evaluated. In order to show this equivalence we prove a version of the Morse's theorem for a manifold with piecewise smooth boundary. These results on the tail probabilities are generalizations of those of Takemura and Kuriki (1997), where M was assumed to be convex.