Intersection and proximity of processes of flats

TL;DR

This paper studies the geometric and probabilistic properties of processes involving random flats in Euclidean space, focusing on intersection and proximity processes, and introduces new stability results and limit theorems.

Contribution

It provides new descriptions of intensity measures, stability results, and limit theorems for processes of flats, expanding understanding of their geometric and probabilistic structure.

Findings

Derived new formulas for intensity measures of flat processes.

Established stability and uniqueness results for these processes.

Proved limit theorems for functionals of proximity processes.

Abstract

Weakly stationary random processes of $k$-dimensional affine subspaces (flats) in $\mathbb{R}^n$ are considered. If $2k\geq n$, 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.