Excursion probability of certain non-centered smooth Gaussian random fields

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Let X={X(t),t∈T} be a non-centered, unit-variance, smooth Gaussian random field indexed on some parameter space T, and let Au(X,T)={t∈T:X(t)≥u} be the excursion set. It is shown that, as u→∞, the excursion probability P{supt∈TX(t)≥u} can be approximated by the expected Euler characteristic of Au(X,T), denoted by E{χ(Au(X,T))}, such that the error is super-exponentially small. The explicit formulae for E{χ(Au(X,T))} are also derived for two cases: (i) T is a rectangle and X−EX is stationary; (ii) T is an N-dimensional sphere and X−EX is isotropic.