Poisson point process convergence and extreme values in stochastic geometry

TL;DR

This paper demonstrates that certain functionals of Poisson point processes in stochastic geometry converge to Poisson processes under rescaling, leading to Weibull limit laws for extreme values, with applications to Voronoi tessellations, simplices, and flats.

Contribution

It establishes general conditions under which functionals of Poisson and binomial processes converge to Poisson processes, extending extreme value theory in stochastic geometry.

Findings

Poisson process convergence for geometric functionals

Weibull limit laws for extreme values

Applications to Voronoi cells, simplices, and flats

Abstract

Let $\eta_t$ be a Poisson point process with intensity measure $t\mu$, $t>0$, over a Borel space $\mathbb{X}$, where $\mu$ is a fixed measure. Another point process $\xi_t$ on the real line is constructed by applying a symmetric function $f$ to every $k$-tuple of distinct points of $\eta_t$. It is shown that $\xi_t$ behaves after appropriate rescaling like a Poisson point process, as $t\to\infty$, under suitable conditions on $\eta_t$ and $f$. This also implies Weibull limit theorems for related extreme values. The result is then applied to investigate problems arising in stochastic geometry, including small cells in Voronoi tessellations, random simplices generated by non-stationary hyperplane processes, triangular counts with angular constraints and non-intersecting $k$-flats. Similar results are derived if the underlying Poisson point process is replaced by a binomial point process.