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Poisson limits for pairwise and area interaction point processes

Part of: Limit theorems Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

S. Rao Jammalamadaka  and
Mathew D. Penrose
S. Rao Jammalamadaka*
Affiliation:
University of California
Mathew D. Penrose*
Affiliation:
University of Durham
*
∗ Postal address: Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106, USA. Email address: rao@pstat.ucsb.edu
∗∗ 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

Suppose n particles xi in a region of the plane (possibly representing biological individuals such as trees or smaller organisms) have a joint density proportional to exp{-∑i<jϕ(n(xi-xj))}, with ℝd; a specified function of compact support. We obtain a Poisson process limit for the collection of rescaled interparticle distances as n becomes large. We give corresponding results for the case of several types of particles, representing different species, and also for the area-interaction (Widom-Rowlinson) point process of interpenetrating spheres.

Keywords

Area-interaction processGibbs distributionlimit lawspoint processspatial statisticsU-statistics

MSC classification

Primary: 60G55: Point processes
Secondary: 60F05: Central limit and other weak theorems 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 32 , Issue 1 , March 2000 , pp. 75 - 85
Copyright
Copyright © Applied Probability Trust 2000 

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