STICKY CENTRAL LIMIT THEOREMS ON OPEN BOOKS

• 成果类型: Article
• 署名作者: Hotz, Thomas; Huckemann, Stephan; Le, Huiling; Marron, J. S.; Mattingly, Jonathan C.; Miller, Ezra; Nolen, James; Owen, Megan; Patrangenaru, Vic; Skwerer, Sean
• 署名单位: Technische Universitat Ilmenau; University of Gottingen; University of Nottingham; University of North Carolina; University of North Carolina Chapel Hill; Duke University; University of Waterloo; State University System of Florida; Florida State University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/12-AAP899
• 发表日期: 2013
• 页码: 2238-2258
• 关键词: extrinsic sample means; MANIFOLDS; matrices; trees; SPACE

摘要

Given a probability distribution on an open book (a metric space obtained by gluing a disjoint union of copies of a half-space along their boundary hyperplanes), we define a precise concept of when the Frechet mean (barycenter) is sticky. This nonclassical phenomenon is quantified by a law of large numbers (LLN) stating that the empirical mean eventually almost surely lies on the (codimension 1 and hence measure 0) spine that is the glued hyperplane, and a central limit theorem (CLT) stating that the limiting distribution is Gaussian and supported on the spine. We also state versions of the LLN and CLT for the cases where the mean is nonsticky (i.e., not lying on the spine) and partly sticky (i.e., is, on the spine but not sticky).