Stochastic Analysis for Poisson Point Processes pp 255-294

Part of the Bocconi & Springer Series book series (BS, volume 7)

Poisson Point Process Convergence and Extreme Values in Stochastic Geometry

Chapter

Abstract

Let ηt 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 ξt on the real line is constructed by applying a symmetric function f to every k-tuple of distinct points of ηt. It is shown that ξt behaves after appropriate rescaling like a Poisson point process, as t → , under suitable conditions on ηt 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.

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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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