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Distances Between Poisson k -Flats

Methodology and Computing in Applied Probability volume 16, pages 311–329 (2014)Cite this article

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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Author notes

  1. Christoph Thäle

    Present address: Ruhr University Bochum, Raum NA 3/68, Universitätsstraße 150, 44801, Bochum, Germany

Authors and Affiliations

  1. Fachbereich Mathematik/Informatik, Universität Osnabrück, Albrechtstraße 28a, 49076, Osnabrück, Germany

    Matthias Schulte & Christoph Thäle

Authors
  1. Matthias Schulte
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  2. Christoph Thäle
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Correspondence to Christoph Thäle.

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Schulte, M., Thäle, C. Distances Between Poisson k -Flats . Methodol Comput Appl Probab 16, 311–329 (2014). https://doi.org/10.1007/s11009-012-9319-2

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  • DOI: https://doi.org/10.1007/s11009-012-9319-2

Keywords

  • Central limit theorem
  • Chaos decomposition
  • Extreme values
  • Limit theorems
  • Poisson flat process
  • Poisson point process
  • Poisson U-statistic
  • Stochastic geometry
  • Wiener–Itô integral

AMS 2000 Subject Classification

  • 60D05
  • 60F05
  • 60G55
  • 60H07