THE EXPECTED EULER CHARACTERISTIC APPROXIMATION TO EXCURSION PROBABILITIES OF GAUSSIAN VECTOR FIELDS

• 成果类型: Article
• 署名作者: Cheng, Dan; Xiao, Yimin
• 署名单位: Arizona State University; Arizona State University-Tempe; Michigan State University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2102
• 发表日期: 2024
• 页码: 5664-5693
• 关键词: extremes; asymptotics

摘要

Let { (X (t), Y(s)) : t is an element of T, s is an element of S } be an R2-valued, centered, unit-variance smooth Gaussian vector field, where T and S are compact rectangles in the Euclidean space. It is shown that, as u -> infinity , the joint excursion probability P{supt is an element of T X (t) >= u, sup s is an element of S Y(s) >= u } can be approximated by E{chi (Au)}, the expected Euler characteristic of the excursion set Au = { (t, s) is an element of T x S : X (t) >= u, Y(s) >= 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.