On the equivalence of the tube and Euler characteristic methods for the distribution of the maximum of Gaussian fields over piecewise smooth domains

• 成果类型: Article
• 署名作者: Takemura, A; Kuriki, S
• 署名单位: University of Tokyo; Research Organization of Information & Systems (ROIS); Institute of Statistical Mathematics (ISM) - Japan
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• 发表日期: 2002
• 页码: 768-796
• 关键词: excursion sets; (chi)over-bar(2) distribution; tail probabilities; INEQUALITY; regression; BOUNDARY; weights; volume; cone

摘要

Consider a Gaussian random field with a finite Karhunen-Loeve expansion of the form Z(u) = Sigma(i=1)(n) u(i)z(i), where z(i), i = 1,..., n, are independent standard normal variables and u = (u(1),..., u(n))' ranges over an index set M, which is a subset of the unit sphere Sn-1 in R-n. Under a very general assumption that M is a manifold with a piecewise smooth boundary, we prove the validity and the equivalence of two currently available methods for obtaining the asymptotic expansion of the tail probability of the maximum of Z(u). One is the tube method, where the volume of the tube around the index set M is evaluated. The other is the Euler characteristic method, where the expectation for the Euler characteristic of the excursion set is evaluated. General discussion on this equivalence was given in a recent paper by R. J. Adler. In order to show the equivalence we prove a version of the Morse theorem for a manifold with a piecewise smooth boundary.