THE MEAN EULER CHARACTERISTIC AND EXCURSION PROBABILITY OF GAUSSIAN RANDOM FIELDS WITH STATIONARY INCREMENTS

• 成果类型: Article
• 署名作者: Cheng, Dan; Xiao, Yimin
• 署名单位: North Carolina State University; Michigan State University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/15-AAP1101
• 发表日期: 2016
• 页码: 722-759
• 关键词: maximum; tube

摘要

Let X = {X (t), t is an element of R-N} be a centered Gaussian random field with stationary increments and X (0) = 0. For any compact rectangle T subset of R-N and u is an element of R, denote by A(u) = {t is an element of T : X (t) >= u} the excursion set. Under X(center dot) is an element of C-2(R-N) and certain regularity conditions, the mean Euler characteristic of A(u), denoted by E{phi(A(u))}, is derived. By applying the Rice method, it is shown that, as u -> infinity, the excursion probability P{sup(t is an element of T) X (t) >= u) can be approximated by E{phi(A(u))} such that the error is exponentially smaller than E{phi(A(u))}. This verifies the expected Euler characteristic heuristic for a large class of Gaussian random fields with stationary increments.