Intersection and proximity of processes of flats

Abstract

Weakly stationary random processes of $k$-dimensional affine subspaces (flats) in ${\mathbb{R}}^n$ are considered. If $2k\ge n$, then intersection processes are investigated, while in the complementary case $2k<n$ a proximity process is introduced. The intensity measures of these processes are described in terms of parameters of the underlying $k$-flat process. By a translation into geometric parameters of associated zonoids and by means of integral transformations, several new uniqueness and stability results for these processes of flats are derived. They rely on a combination of known and novel estimates for area measures of zonoids, which are also developed in the paper. Finally, an asymptotic second-order analysis as well as central and non-central limit theorems for length-power direction functionals of proximity processes derived from stationary Poisson $k$-flat process complement earlier works for intersection processes.