Distances Between Poisson k
-Flats |
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Authors: | Matthias Schulte Christoph Thäle |
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Institution: | 1. Fachbereich Mathematik/Informatik, Universit?t Osnabrück, Albrechtstra?e 28a, 49076, Osnabrück, Germany 2. Ruhr University Bochum, Raum NA 3/68, Universit?tsstra?e 150, 44801, Bochum, Germany
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Abstract: | The distances between flats of a Poisson k-flat process in the d-dimensional Euclidean space with k?<?d/2 are discussed. Continuing an approach originally due to Rolf Schneider, the number of pairs of flats having distance less than a given threshold and midpoint in a fixed compact and convex set is considered. For a family of increasing convex subsets, the asymptotic variance is computed and a central limit theorem with an explicit rate of convergence is proven. Moreover, the asymptotic distribution of the m-th smallest distance between two flats is investigated and it is shown that the ordered distances form asymptotically after suitable rescaling an inhomogeneous Poisson point process on the positive real half-axis. A similar result with a homogeneous limiting process is derived for distances around a fixed, strictly positive value. Our proofs rely on recent findings based on the Wiener–Itô chaos decomposition and the Malliavin–Stein method. |
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