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 $u \to \infty$, the excursion probability $\mathbb{P}{\sup_{t\in T} X(t) \geq u}$ can be accurately approximated by $\mathbb{E}{\chi(A_u)}$ such that the error decays at a super-exponential rate. Here, $A_u = {t\in T: X(t)\geq u}$ represents the excursion set above $u$, and $\mathbb{E}{\chi(A_u)}$ is the expectation of its Euler characteristic $\chi(A_u)$. 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.