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 $\mathbb{P}\{{\sup }_{t\in T}X(t)\ge u,{\sup }_{s\in S}Y(s)\ge u\}$ 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.