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

Abstract

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