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

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    Let ηt\eta_t be a Poisson point process of intensity t≥1t\geq 1 on some state space \Y and ff be a non-negative symmetric function on \Y^k for some k≥1k\geq 1. Applying ff to all kk-tuples of distinct points of ηt\eta_t generates a point process ξt\xi_t on the positive real-half axis. The scaling limit of ξt\xi_t as tt tends to infinity is shown to be a Poisson point process with explicitly known intensity measure. From this, a limit theorem for the the mm-th smallest point of ξt\xi_t is concluded. This is strengthened by providing a rate of convergence. The technical background includes Wiener-It\^o 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 kk-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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