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 of X exceeding level u. Under certain smoothness and regularity conditions, it is shown that, as u→ ∞, the excursion probability P{ₜ∈ ₜ X(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. 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 E{χ(Aᵤ(X,T))} are also derived for two cases: (i) T is a rectangle and X-E X is stationary; (ii) T is an N-dimensional sphere and X-E X is isotropic.