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Poisson Point Process Convergence and Extreme Values in Stochastic Geometry

  • Matthias SchulteEmail author
  • Christoph Thäle
Chapter
Part of the Bocconi & Springer Series book series (BS, volume 7)

Abstract

Let η t be a Poisson point process with intensity measure , t > 0, over a Borel space \(\mathbb{X}\), where μ is a fixed measure. Another point process ξ t on the real line is constructed by applying a symmetric function f to every k-tuple of distinct points of η t . It is shown that ξ t behaves after appropriate rescaling like a Poisson point process, as t → , under suitable conditions on η t and f. This also implies Weibull limit theorems for related extreme values. The result is then applied to investigate problems arising in stochastic geometry, including small cells in Voronoi tessellations, random simplices generated by non-stationary hyperplane processes, triangular counts with angular constraints, and non-intersecting k-flats. Similar results are derived if the underlying Poisson point process is replaced by a binomial point process.

Keywords

Point Process Convex Body Poisson Point Process Voronoi Tessellation Poisson Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Karlsruhe Institute of TechnologyInstitute of StochasticsKarlsruheGermany
  2. 2.Faculty of MathematicsRuhr University BochumBochumGermany

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