The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments

Abstract

Let $X=\{X(t),t\in\mathbb{R}^{N}\}$ be a centered Gaussian random field with stationary increments and $X(0)=0$. For any compact rectangle $T\subset\mathbb{R}^{N}$ and $u\in\mathbb{R}$, denote by $A_{u}=\{t\in T:X(t)\geq u\}$ the excursion set. Under $X(\cdot)\in C^{2}(\mathbb{R}^{N})$ and certain regularity conditions, the mean Euler characteristic of $A_{u}$, denoted by $\mathbb{E}\{\varphi(A_{u})\}$, is derived. By applying the Rice method, it is shown that, as $u\to\infty$, the excursion probability $\mathbb{P}\{\sup_{t\in T}X(t)\geq u\}$ can be approximated by $\mathbb{E}\{\varphi(A_{u})\}$ such that the error is exponentially smaller than $\mathbb{E}\{\varphi(A_{u})\}$. This verifies the expected Euler characteristic heuristic for a large class of Gaussian random fields with stationary increments.