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Let $\eta_t$ be a Poisson point process of intensity $t\geq 1$ on some statespace $\Y$ 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 $\eta_t$generates a point process $\xi_t$ on the positive real-half axis. The scalinglimit of $\xi_t$ as $t$ tends to infinity is shown to be a Poisson pointprocess with explicitly known intensity measure. From this, a limit theorem forthe the $m$-th smallest point of $\xi_t$ is concluded. This is strengthened byproviding a rate of convergence. The technical background includes Wiener-It\^ochaos decompositions and the Malliavin calculus of variations on the Poissonspace as well as the Chen-Stein method for Poisson approximation. The generalresult is accompanied by a number of examples from geometric probability andstochastic geometry, such as Poisson $k$-flats, Poisson random polytopes,random geometric graphs and random simplices. They are obtained by combiningthe general limit theorem with tools from convex and integral geometry.

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