GAUSSIAN PROCESSES, KINEMATIC FORMULAE AND POINCARE'S LIMIT

• 成果类型: Article
• 署名作者: Taylor, Jonathan E.; Adler, Robert J.
• 署名单位: Stanford University; Technion Israel Institute of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/08-AOP439
• 发表日期: 2009
• 页码: 1459-1482
• 关键词: excursion sets; random-field; unknown location; maxima; chi(2); scale

摘要

We consider vector valued, unit variance Gaussian processes defined over stratified manifolds and the geometry of their excursion sets. In particular, we develop an explicit formula for the expectation of all the Lipschitz-Killing curvatures of these sets. Whereas our motivation is primarily probabilistic, with statistical applications in the background, this formula has also an interpretation as a version of the classic kinematic fundamental formula of integral geometry. All of these aspects are developed in the paper. Particularly novel is the method of proof, which is based on a an approximation to the canonical Gaussian process on the n-sphere. The n -> infinity limit, which gives the final result, is handled via recent extensions of the classic Poincare limit theorem.