The Annals of Applied Probability

Sticky central limit theorems on open books

Thomas Hotz, Stephan Huckemann, Huiling Le, J. S. Marron, Jonathan C. Mattingly, Ezra Miller, James Nolen, Megan Owen, Vic Patrangenaru, and Sean Skwerer

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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 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).

Article information

Source
Ann. Appl. Probab., Volume 23, Number 6 (2013), 2238-2258.

Dates
First available in Project Euclid: 22 October 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1382447687

Digital Object Identifier
doi:10.1214/12-AAP899

Mathematical Reviews number (MathSciNet)
MR3127934

Zentralblatt MATH identifier
1293.60006

Subjects
Primary: 60B99: None of the above, but in this section 60F05: Central limit and other weak theorems

Keywords
Fréchet mean central limit theorem law of large numbers stratified space nonpositive curvature

Citation

Hotz, Thomas; Huckemann, Stephan; Le, Huiling; Marron, J. S.; Mattingly, Jonathan C.; Miller, Ezra; Nolen, James; Owen, Megan; Patrangenaru, Vic; Skwerer, Sean. Sticky central limit theorems on open books. Ann. Appl. Probab. 23 (2013), no. 6, 2238--2258. doi:10.1214/12-AAP899. https://projecteuclid.org/euclid.aoap/1382447687


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