The scaling limit of Poisson-driven order statistics with applications in geometric probability |
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Authors: | Matthias Schulte Christoph Thäle |
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Institution: | Institut für Mathematik der Universität Osnabrück, Albrechtstraße 28a, 49076 Osnabrück, Germany |
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Abstract: | Let ηt be a Poisson point process of intensity t≥1 on some state space Y and let f be a non-negative symmetric function on Yk for some k≥1. Applying f to all k-tuples of distinct points of ηt generates a point process ξt on the positive real half-axis. The scaling limit of ξt 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 m-th smallest point of ξt 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 k-flats, 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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Keywords: | primary 60F17 60D05 62G32 secondary 60G55 60H07 |
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