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Poisson Point Process Convergence and Extreme Values in Stochastic Geometry

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Book cover Stochastic Analysis for Poisson Point Processes

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

Abstract

Let η t be a Poisson point process with intensity measure tμ, 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.

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Author information

Authors and Affiliations

  1. Karlsruhe Institute of Technology, Institute of Stochastics, 76128, Karlsruhe, Germany

    Matthias Schulte

  2. Faculty of Mathematics, Ruhr University Bochum, 44780, Bochum, Germany

    Christoph Thäle

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

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Editors and Affiliations

  1. Unité de Recherche en Mathématiques, Université du Luxembourg, Luxembourg, Luxembourg

    Giovanni Peccati

  2. Institut für Mathematik, Osnabrück University, Osnabrück, Germany

    Matthias Reitzner

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Schulte, M., Thäle, C. (2016). Poisson Point Process Convergence and Extreme Values in Stochastic Geometry. In: Peccati, G., Reitzner, M. (eds) Stochastic Analysis for Poisson Point Processes. Bocconi & Springer Series, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-319-05233-5_8

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