Intersections of Poisson k-flats in hyperbolic space: completing the picture

In recent years there has been a lot of interest in the study of isometry invariant Poisson processes of $k$-flats in $d$-dimensional hyperbolic space ${\mathbb{H}}^d$, for $0\le k\le d-1$. A phenomenon that has no counterpart in euclidean geometry arises in the investigation of the total $k$-dimensional volume $F_r$ of the process inside a spherical observation window $B_r$ of radius $r$ when one lets $r$ tend to infinity. While $F_r$ is asymptotically normally distributed for $2k\le d+1$, it has been shown to obey a nonstandard central limit theorem for $2k>d+1$. The intersection process of order $m$, for $d-m(d-k)\ge 0$, of the original process $\eta$ consists of all intersections of distinct flats $E_1,\dots ,E_m\in \eta$ with $\dim (E_1\cap \dots \cap E_m)=d-m(d-k)$. For this intersection process, the total $d-m(d-k)$-dimensional volume $F_r^{(m)}$ of the process in $B_r$, again as $r\to \mathrm{\infty }$, is of particular interest. For $2k\le d+1$ it has been shown that $F_r^{(m)}$ is again asymptotically normally distributed. For $m\ge 2$, the limit is so far unknown, although it has been shown for certain $d$ and $k$ that it cannot be a normal distribution. We determine the limit distribution for all values of $d,k,m$. In addition, we establish explicit rates of convergence in the Kolmogorov distance and discuss properties of the limit distribution. Furthermore we show that the asymptotic covariance matrix of the vector $(F_r^{(1)},\dots ,F_r^{(m)}{)}^{\mathrm{\top }}$ has full rank when $2k<d+1$ and rank one when $2k\ge d+1$.