Year: 2013
Type: article
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).
DOI: 10.1214/12-aap899
Authors: Thomas Hotz, Sean Skwerer, Stephan Huckemann, Huiling Le, J. S. Marron, Jonathan C. Mattingly, Ezra Miller, James Nolen, Megan Owen, Vic Patrangenaru
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
Awards: Combinatorics in geometry and algebra with applications to the natural sciences, Collaborative Research: Tree Structured Object Oriented Data Analysis, AMC-SS: Analysis of Fluctuations for PDEs with Random Coefficients, G4818080411, G5186959748, FRG: Collaborative Research: Stochastics and Dynamics: Asymptotic problems, Collaborative Research: Nonparametric Theory on Manifolds of Shapes and Images, with Applications to Biology, Medical Imaging and Machine Vision., Collaborative Research: New Directions in Nonparametric Inference on Manifolds with Applications to Shapes and Images
OpenAlex ID: W2018356552