The expected Euler characteristic approximation to excursion probabilities of Gaussian vector fields

Let {(X(t),Y(s)):t∈T,s∈S} be an R2-valued, centered, unit-variance smooth Gaussian vector field, where T and S are compact rectangles in the Euclidean space. It is shown that, as u→∞, the joint excursion probability P{supt∈TX(t)≥u,sups∈SY(s)≥u} can be approximated by E{χ(Au)}, the expected Euler characteristic of the excursion set Au={(t,s)∈T×S:X(t)≥u,Y(s)≥u}, such that the error is super-exponentially small. This verifies the expected Euler characteristic heuristic (cf. Taylor, Takemura and Alder (2005), Alder and Taylor (2007)) for a large class of smooth Gaussian vector fields.