Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space?

Abstract

Poisson processes in the space of $$(d-1)$$ ( d - 1 ) -dimensional totally geodesic subspaces (hyperplanes) in a d-dimensional hyperbolic space of constant curvature $$-1$$ - 1 are studied. The k-dimensional Hausdorff measure of their k-skeleton is considered. Explicit formulas for first- and second-order quantities restricted to bounded observation windows are obtained. The central limit problem for the k-dimensional Hausdorff measure of the k-skeleton is approached in two different set-ups: (i) for a fixed window and growing intensities, and (ii) for fixed intensity and growing spherical windows. While in case (i) the central limit theorem is valid for all $$d\ge 2$$ d ≥ 2 , it is shown that in case (ii) the central limit theorem holds for $$d\in \{2,3\}$$ d ∈ { 2 , 3 } and fails if $$d\ge 4$$ d ≥ 4 and $$k=d-1$$ k = d - 1 or if $$d\ge 7$$ d ≥ 7 and for general k. Also rates of convergence are studied and multivariate central limit theorems are obtained. Moreover, the situation in which the intensity and the spherical window are growing simultaneously is discussed. In the background are the Malliavin–Stein method for normal approximation and the combinatorial moment structure of Poisson U-statistics as well as tools from hyperbolic integral geometry.

The Typical Cell of a Voronoi Tessellation on the Sphere

“…We would like to point out that our paper continues a recent line of research in stochastic geometry which focuses on the study of non-Euclidean geometric random structures. As examples we mention the studies of random convex hulls in spherical convex bodies or on half-spheres [3,5,21,23], the results on random tessellations by great hyperspheres [1,16,17,27], the central and non-central limit theorems for Poisson hyperplanes in hyperbolic spaces [15], the papers [12,18] on splitting tessellations on the sphere, the asymptotic investigation of Voronoi tessellations on general Riemannian manifolds [9], and the general limit theory for stabilizing functionals of point processes in manifolds [32].…”

Random inscribed polytopes in projective geometries

“…This continues a recent trend in stochastic geometry of generalizing known results to the non-Euclidean setting, and in particular to spherical and hyperbolic geometry, see e.g. [5,6,24,28,29,[32][33][34][35][36][37][38].…”

Fluctuations of $λ$-geodesic Poisson hyperplanes in hyperbolic space

“…Examples include the study of random hyperbolic Voronoi tessellations [3,17,16], hyperbolic random polytopes [4,5], the hyperbolic Boolean model [2,36,37], hyperbolic Poisson cylinder processes [8] or hyperbolic random geometric graphs [14,26]. Most closely related to the present paper are the investigations in [19] about Poisson processes of hyperplanes, that is, totally geodesic (d−1)-dimensional submanifolds, in d-dimensional hyperbolic spaces. While first-order properties of geometric functionals associated with such process are independent of the curvature of the underlying space, it turned out that second-order parameters and also the accompanying central limit theory are rather sensitive to curvature.…”