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December 2013 Sticky central limit theorems on open books
Thomas Hotz, Sean Skwerer, Stephan Huckemann, Huiling Le, J. S. Marron, Jonathan C. Mattingly, Ezra Miller, James Nolen, Megan Owen, Vic Patrangenaru
Ann. Appl. Probab. 23(6): 2238-2258 (December 2013). DOI: 10.1214/12-AAP899


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 Fréchet 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).


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Thomas Hotz. Sean Skwerer. Stephan Huckemann. Huiling Le. J. S. Marron. Jonathan C. Mattingly. Ezra Miller. James Nolen. Megan Owen. Vic Patrangenaru. "Sticky central limit theorems on open books." Ann. Appl. Probab. 23 (6) 2238 - 2258, December 2013.


Published: December 2013
First available in Project Euclid: 22 October 2013

zbMATH: 1293.60006
MathSciNet: MR3127934
Digital Object Identifier: 10.1214/12-AAP899

Primary: 60B99 , 60F05

Keywords: central limit theorem , Fréchet mean , Law of Large Numbers , nonpositive curvature , stratified space

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.23 • No. 6 • December 2013
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