Electronic Journal of Probability

Sticky central limit theorems at isolated hyperbolic planar singularities

Stephan Huckemann, Jonathan Mattingly, Ezra Miller, and James Nolen

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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.

Article information

Electron. J. Probab. Volume 20 (2015), paper no. 78, 34 pp.

Accepted: 21 July 2015
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

This work is licensed under a Creative Commons Attribution 3.0 License.


Huckemann, Stephan; Mattingly, Jonathan; Miller, Ezra; Nolen, James. Sticky central limit theorems at isolated hyperbolic planar singularities. Electron. J. Probab. 20 (2015), paper no. 78, 34 pp. doi:10.1214/EJP.v20-3887. https://projecteuclid.org/euclid.ejp/1465067184.

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