Intersections of Poisson [formula omitted]-flats in constant curvature spaces

Abstract

Poisson processes in the space of k-dimensional totally geodesic subspaces (k-flats) in a d-dimensional standard space of constant curvature κ∈{−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)≥0. For κ∈{−1,0} the problem is also approached in the set-up where the intensity is fixed and r tends to infinity. Again, if 2k≤d+1 a central limit theorem is shown for all possible values of m. However, while for κ=0 asymptotic normality still holds if 2k>d+1, we prove for κ=−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 κ∈{−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.