The scaling limit of Poisson-driven order statistics with applications in geometric probability

Let $η_t$ be a Poisson point process of intensity $t\geq 1$ on some state space $\Y$ and $f$ be a non-negative symmetric function on $\Y^k$ for some $k\geq 1$. Applying $f$ to all $k$-tuples of distinct points of $η_t$ generates a point process $ξ_t$ on the positive real-half axis. The scaling limit of $ξ_t$ as $t$ tends to infinity is shown to be a Poisson point process with explicitly known intensity measure. From this, a limit theorem for the the $m$-th smallest point of $ξ_t$ is concluded. This is strengthened by providing a rate of convergence. The technical background includes Wiener-Itô chaos decompositions and the Malliavin calculus of variations on the Poisson space as well as the Chen-Stein method for Poisson approximation. The general result is accompanied by a number of examples from geometric probability and stochastic geometry, such as Poisson $k$-flats, Poisson random polytopes, random geometric graphs and random simplices. They are obtained by combining the general limit theorem with tools from convex and integral geometry.

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