Excursion Probability of Certain Non-centered Smooth Gaussian Random Fields

Abstract: Let $X = {X(t): t\in T }$ be a non-centered, unit-variance, smooth Gaussian random field indexed on some parameter space $T$, and let $A_u(X,T) = {t\in T: X(t)\geq u}$ be the excursion set of $X$ exceeding level $u$. Under certain smoothness and regularity conditions, it is shown that, as $u\to \infty$, the excursion probability $\mathbb{P}{\sup_{t\in T} X(t)\ge u }$ can be approximated by the expected Euler characteristic of $A_u(X,T)$, denoted by $\mathbb{E}{\chi(A_u(X,T))}$, such that the error is super-exponentially small. This verifies the expected Euler characteristic heuristic for a large class of non-centered smooth Gaussian random fields and provides a much more accurate approximation compared with those existing results by the double sum method. The explicit formulae for $\mathbb{E}{\chi(A_u(X,T))}$ are also derived for two cases: (i) $T$ is a rectangle and $X-\mathbb{E} X$ is stationary; (ii) $T$ is an $N$-dimensional sphere and $X-\mathbb{E} X$ is isotropic.