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

TL;DR

This paper investigates whether a central limit theorem applies to the measure of the $k$-skeleton of Poisson hyperplanes in hyperbolic space, revealing dimension-dependent validity and convergence rates.

Contribution

It provides explicit formulas, analyzes CLT validity under different growth regimes, and explores convergence rates for Poisson hyperplane processes in hyperbolic geometry.

Findings

CLT holds for fixed windows with growing intensity in all dimensions

CLT valid for $d=2,3$ but fails for higher dimensions in certain cases

Rates of convergence and multivariate CLTs are established

Abstract

Poisson processes in the space of $(d-1)$-dimensional totally geodesic subspaces (hyperplanes) in a $d$-dimensional hyperbolic space of constant curvature $-1$ 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 $d\geq 2$, it is shown that in case (ii) the central limit theorem holds for $d\in\{2,3\}$ and fails if $d\geq 4$ and $k=d-1$ or if $d\geq 7$ 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.