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Poisson-saddlepoint approximation for Gibbs point processes with infinite-order interaction: in memory of Peter Hall

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

Published online by Cambridge University Press:  30 November 2017

Adrian Baddeley  and
Gopalan Nair
Adrian Baddeley*
Affiliation:
Curtin University
Gopalan Nair*
Affiliation:
University of Western Australia
*
* Postal address: Department of Mathematics & Statistics, Curtin University, GPO Box U1987, Perth, WA 6845, Australia. Email address: adrian.baddeley@curtin.edu.au
** Postal address: School of Mathematics & Statistics, University of Western Australia, 35 Stirling Highway, Nedlands, WA 6009, Australia. Email address: gopalan.nair@uwa.edu.au
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Abstract

We develop a computational approximation to the intensity of a Gibbs spatial point process having interactions of any order. Limit theorems from stochastic geometry, and small-sample probabilities estimated once and for all by an extensive simulation study, are combined with scaling properties to form an approximation to the moment generating function of the sufficient statistic under a Poisson process. The approximate intensity is obtained as the solution of a self-consistency equation.

Keywords

Area-interaction processcoverage processGeorgii–Nguyen–Zessin formulaGeyer saturation processGibbs point processPoisson line tessellation

MSC classification

Primary: 60G55: Point processes
Secondary: 60D05: Geometric probability and stochastic geometry 62E17: Approximations to distributions (nonasymptotic)
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 4 , December 2017 , pp. 1008 - 1026
Copyright
Copyright © Applied Probability Trust 2017 

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