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Focusing of the scan statistic and geometric clique number

Part of: Geometric probability and stochastic geometry Stochastic processes Graph theory

Published online by Cambridge University Press:  01 July 2016

Mathew D. Penrose
Mathew D. Penrose*
Affiliation:
University of Durham
*
∗ Postal address: Department of Mathematical Sciences, University of Durham, South Road, Durham DH1 3LE, UK. Email address: mathew.penrose@durham.ac.uk
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Abstract

Given sets C and R in d-dimensional space, take a constant intensity Poisson point process on R; the associated scan statistic S is the maximum number of Poisson points in any translate of C. As R becomes large with C fixed, bounded and open but otherwise arbitrary, the distribution of S becomes concentrated on at most two adjacent integers. A similar result holds when the underlying Poisson process is replaced by a binomial point process, and these results can be extended to a large class of nonuniform distributions. Also, similar results hold for other finite-range scanning schemes such as the clique number of a geometric graph.

Keywords

Extremesrandom graphasymptotically quasi-deterministicChen-Stein

MSC classification

Primary: 60G70: Extreme value theory; extremal processes 60D05: Geometric probability and stochastic geometry 05C80: Random graphs
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 34 , Issue 4 , December 2002 , pp. 739 - 753
Copyright
Copyright © Applied Probability Trust 2002 

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