Poisson point process convergence and extreme values in stochastic geometry

Let $η_t$ be a Poisson point process with intensity measure $tμ$, $t>0$, over a Borel space $\mathbb{X}$, where $μ$ is a fixed measure. Another point process $ξ_t$ on the real line is constructed by applying a symmetric function $f$ to every $k$-tuple of distinct points of $η_t$. It is shown that $ξ_t$ behaves after appropriate rescaling like a Poisson point process, as $t\to\infty$, under suitable conditions on $η_t$ 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.

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