The expected Euler characteristic approximation to excursion probabilities of smooth Gaussian random fields with general variance functions

Abstract

Consider a centered smooth Gaussian random field $\{X(t),t\in T\}$ with a general (nonconstant) variance function. In this work, we demonstrate that as $u\to \mathrm{\infty }$, the excursion probability $\mathbb{P}\{{sup}_{t\in T}X(t)\ge u\}$ can be accurately approximated by $\mathbb{E}\{\mathit{\chi }(A_u)\}$ such that the error decays at a super-exponential rate. Here, $A_u=\{t\in T:X(t)\ge u\}$ represents the excursion set above u, and $\mathbb{E}\{\mathit{\chi }(A_u)\}$ is the expectation of its Euler characteristic $\mathit{\chi }(A_u)$. This result substantiates the expected Euler characteristic heuristic for a broad class of smooth Gaussian random fields with diverse covariance structures. In addition, we employ the Laplace method to derive explicit approximations to the excursion probabilities.