Sticky central limit theorems on open books
Abstract
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 Fr\'{e}chet 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 and hence measure ) 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).
Keywords
Cite
@article{arxiv.1202.4267,
title = {Sticky central limit theorems on open books},
author = {Thomas Hotz and Sean Skwerer and Stephan Huckemann and Huiling Le and J. S. Marron and Jonathan C. Mattingly and Ezra Miller and James Nolen and Megan Owen and Vic Patrangenaru},
journal= {arXiv preprint arXiv:1202.4267},
year = {2013}
}
Comments
Published in at http://dx.doi.org/10.1214/12-AAP899 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)