## Electronic Journal of Probability

### Sticky central limit theorems at isolated hyperbolic planar singularities

#### 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.

#### Article information

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

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

https://projecteuclid.org/euclid.ejp/1465067184

Digital Object Identifier
doi:10.1214/EJP.v20-3887

Mathematical Reviews number (MathSciNet)
MR3371437

Zentralblatt MATH identifier
1327.60028

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

Rights

#### Citation

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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