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

Abstract: Let ${(X(t), Y(s)): t\in T, s\in S}$ be an $\mathbb{R}^2$-valued, centered, unit-variance smooth Gaussian vector field, where $T$ and $S$ are compact rectangles in $\mathbb{R}^N$. It is shown that, as $u\to \infty$, the joint excursion probability $\mathbb{P} {\sup_{t\in T} X(t) \geq u, \sup_{s\in S} Y(s) \geq u }$ can be approximated by $\mathbb{E}{\chi(A_u)}$, the expected Euler characteristic of the excursion set $A_u={(t,s)\in T\times S: X(t) \ge u, Y(s) \ge u}$, 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.