Intersections of Poisson \( k \)-flats in constant curvature spaces

• Published in: arXiv.org 2023-02
• Main Authors: Betken, Carina, Hug, Daniel, Thäle, Christoph
• Format: Article
• Language: English
• Subjects: Asymptotic methods; Asymptotic properties; Curvature; Infinity; Intersections; Normality; Poisson density functions; Subspaces
• ISSN: 2331-8422

Poisson processes in the space of \(k\)-dimensional totally geodesic subspaces (\(k\)-flats) in a \(d\)-dimensional standard space of constant curvature \(\kappa\in\{-1,0,1\}\) are studied, whose distributions are invariant under the isometries of the space. We consider the intersection processes of order \(m\) together with their \((d-m(d-k))\)-dimensional Hausdorff measure within a geodesic ball of radius \(r\). Asymptotic normality for fixed \(r\) is shown as the intensity of the underlying Poisson process tends to infinity for all \(m\) satisfying \(d-m(d-k)\geq 0\). For \(\kappa\in\{-1,0\}\) the problem is also approached in the set-up where the intensity is fixed and \(r\) tends to infinity. Again, if \(2k\le d+1\) a central limit theorem is shown for all possible values of \(m\). However, while for \(\kappa=0\) asymptotic normality still holds if \(2k>d+1\), we prove for \(\kappa=-1\) convergence to a non-Gaussian infinitely divisible limit distribution in the special case \(m=1\). The proof of asymptotic normality is based on the analysis of variances and general bounds available from the Malliavin--Stein method. We also show for general \(\kappa\in\{-1,0,1\}\) that, roughly speaking, the variances within a general observation window \(W\) are maximal if and only if \(W\) is a geodesic ball having the same volume as \(W\). Along the way we derive a new integral-geometric formula of Blaschke--Petkantschin type in a standard space of constant curvature.