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arXiv · 1410.6879

Sticky central limit theorems at isolated hyperbolic planar singularities

Stephan Huckemann·Jonathan C. Mattingly·Ezra Miller·James Nolen
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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 $2π$. 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.

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Stephan Huckemann, Jonathan C. Mattingly, Ezra Miller, James Nolen. 2015-07-13. Sticky central limit theorems at isolated hyperbolic planar singularities. https://arxiv.org/abs/1410.6879

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