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.
Highlights
It has recently been observed that large samples from well-behaved probability distributions on metric spaces that are not smooth Riemannian manifolds are sometimes constrained to lie in subsets of low dimension, and that central limit theorems in such cases behave non-classically, with components of limiting distributions supported on thin subsets of the sample space [14, 2, 4]
We prove laws of large numbers (Theorem 1.12; see Section 5 for proofs and more details) as well as central limit theorems (Section 1.4; proofs in Section 6) for Fréchet means of probability distributions (Definitions 1.6 and 1.7) on metric spaces possessing the simplest geometric singularities in codimension 2
In the sticky case this support degenerates in some specified sense already in finite random time (Theorem 1.12)
Summary
It has recently been observed that large samples from well-behaved probability distributions on metric spaces that are not smooth Riemannian manifolds are sometimes constrained to lie in subsets of low dimension, and that central limit theorems in such cases behave non-classically, with components of limiting distributions supported on thin subsets of the sample space [14, 2, 4]. We prove laws of large numbers (Theorem 1.12; see Section 5 for proofs and more details) as well as central limit theorems (Section 1.4; proofs in Section 6) for Fréchet means of probability distributions (Definitions 1.6 and 1.7) on metric spaces possessing the simplest geometric singularities in codimension 2. The simpler nature of an isolated planar singularity, which lacks the global combinatorial complexity of tree space, allows us to discover these boundary components and characterize them by identifying the limit measure as the convex projection of a Gaussian distribution (Theorem 1.14). The greater challenge consists in the partly sticky case; as is the case for the multivariate Central Limit Theorem, as well as for that on manifolds by [10, 11, 6] or on certain stratified spaces by [16], square-integrability is still required.
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