Sticky central limit theorems on open books

TL;DR

This paper establishes sticky central limit theorems for probability measures on open books, showing that empirical means tend to concentrate on the spine and that fluctuations follow a Gaussian distribution supported on it.

Contribution

It introduces a rigorous concept of stickiness for Fréchet means on open books and proves LLN and CLT results describing their asymptotic behavior.

Findings

Empirical means almost surely lie on the spine in the sticky case.

Limiting distribution of the mean is Gaussian and supported on the spine.

Results extend to nonsticky and partly sticky cases.

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 $1$ 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).