Distances between Poisson k-flats

TL;DR

This paper studies the statistical properties of distances between flats in a Poisson flat process, deriving limit theorems and distributional results using advanced probabilistic techniques.

Contribution

It provides new asymptotic distributional results for distances between flats in Poisson processes, including central limit theorems and point process convergence.

Findings

Asymptotic variance of pairwise distances is computed.

Distances form an inhomogeneous Poisson process after rescaling.

Results apply to distances around fixed positive values.

Abstract

The distances between flats of a Poisson $k$-flat process in the $d$-dimensional Euclidean space with $k<d/2$ are discussed. Continuing an approach originally due to Rolf Schneider, the number of pairs of flats having distance less than a given threshold and midpoint in a fixed compact and convex set is considered. For a family of increasing convex subsets, the asymptotic variance is computed and a central limit theorem with an explicit rate of convergence is proven. Moreover, the asymptotic distribution of the $m$-th smallest distance between two flats is investigated and it is shown that the ordered distances form asymptotically after suitable rescaling an inhomogeneous Poisson point process on the positive real axis. A similar result with a homogeneous limiting process is derived for distances around a fixed, strictly positive value. Our proofs rely on recent findings based on the Wiener-It\^o chaos decomposition and the Malliavin-Stein method.