Sticky central limit theorems at isolated hyperbolic planar singularities

TL;DR

This paper establishes the limiting distribution of barycenters for i.i.d. points on hyperbolic cones with large angular spread, revealing three distinct sticky and non-sticky regimes and proposing a topological notion of stickiness.

Contribution

It introduces a comprehensive framework for understanding the asymptotic behavior of barycenters in hyperbolic cone spaces, including new classifications of stickiness and a topological perspective.

Findings

Identifies three regimes: fully sticky, partly sticky, and non-sticky.

Describes the limiting distributions for each regime, including Gaussian and mixed types.

Proposes a topological definition of stickiness applicable to general metric spaces.

Abstract

We derive the limiting distribution of the barycenter $b_n$ of an i.i.d. sample of $n$ random points on a planar cone with angular spread larger than $2\pi$. There are three mutually exclusive possibilities: (i) (fully sticky case) after a finite random time the barycenter is almost surely at the origin; (ii) (partly sticky case) the limiting distribution of $\sqrt{n} b_n$ comprises a point mass at the origin, an open sector of a Gaussian, and the projection of a Gaussian to the sector's bounding rays; or (iii) (nonsticky case) the barycenter stays away from the origin and the renormalized fluctuations have a fully supported limit distribution---usually Gaussian but not always. We conclude with an alternative, topological definition of stickiness that generalizes readily to measures on general metric spaces.