The Mean Euler Characteristic and Excursion Probability of Gaussian Random Fields with Stationary Increments

• Activity Number: 558
• Type: Contributed
• Date/Time: Wednesday, August 6, 2014 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Yimin Xiao *+ and Dan Cheng
• Companies: Michigan State University and
• Keywords: Tail probability ; Supremum of Gaussian fields ; Euler characteristic ; excursion sets ; the Rice formula ; Morse function

Abstract

Let $X = \{X(t), t\in {\mathbb R}^{N} \}$ be a centered Gaussian random field with stationary increments and let $T \subset {\mathbb R}^N$ be a compact rectangle. Under $X(\cdot) \in C^2({\mathbb R}^N)$ and certain additional regularity conditions, the mean Euler characteristic of the excursion set $A_u = \{t\in T: X(t)\geq u\}$, 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.