Intersections of Poisson $k$-flats in constant curvature spaces

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 $d-m(d-k)\ge 0.For\kappa \in -1,0$ 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 $W$. ... mehrAlong the way we derive a new integral-geometric formula of Blaschke--Petkantschin type in a standard space of constant curvature.