Poisson point process convergence and extreme values in stochastic geometry

Schulte, Matthias; Thäle, Christoph (2016). Poisson point process convergence and extreme values in stochastic geometry. In: Peccati, Giovanni; Reitzner, Matthias (eds.) Stochastic Analysis for Poisson Point Processes. Bocconi & Springer Series: Vol. 7 (pp. 255-294). Springer 10.1007/978-3-319-05233-5_8

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Let η t be a Poisson point process with intensity measure tμ , t>0 , over a Borel space 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.

Item Type:

Book Section (Book Chapter)

Division/Institute:

08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematical Statistics and Actuarial Science

UniBE Contributor:

Schulte, Matthias

Subjects:

500 Science > 510 Mathematics

ISSN:

2039-1471

ISBN:

978-3-319-05232-8

Series:

Bocconi & Springer Series

Publisher:

Springer

Language:

English

Submitter:

David Ginsbourger

Date Deposited:

25 Apr 2017 11:25

Last Modified:

25 Apr 2017 11:25

Publisher DOI:

10.1007/978-3-319-05233-5_8

BORIS DOI:

10.7892/boris.93221

URI:

https://boris.unibe.ch/id/eprint/93221

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