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Poisson limits for pairwise and area interaction point processes

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

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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References

[1] Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximations: the Chen–Stein method. Ann. Prob. 7, 9–25.Google Scholar
[2] Baddeley, A. J. and van Lieshout, M. N. M. (1995). Area-interaction point processes. Ann. Inst. Statist. Math. 47, 601–619.Google Scholar
[3] Barbour, A. D. and Eagleson, G. K. (1983). Poisson approximation for some statistics based on exchangeable trials. Adv. Appl. Prob. 15, 585–600.Google Scholar
[4] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
[5] Diggle, P. J., Fiksel, T., Grabarnik, P., Ogata, Y., Stoyan, D. and Tanemura, M. (1994). On parameter estimation for pairwise interaction point processes. Int. Statist. Rev. 62, 99–117.CrossRefGoogle Scholar
[6] Geyer, C. J. and Thompson, E. A. (1995). Annealing Markov chain Monte Carlo with applications to ancestral inference. J. Amer. Statist. Assoc. 90, 909–920.Google Scholar
[7] Jammalamadaka, S. R. and Janson, S. (1986). Limit theorems for a triangular scheme of U-statistics with applications to inter-point distances. Ann. Prob. 14, 1347–1358.Google Scholar
[8] Kallenberg, O. (1973). Characterization and convergence of random measures and point processes, Z. Wahrscheinlichkeitsth. 27, 9–21.CrossRefGoogle Scholar
[9] Møller, J., (1999). Markov chain Monte Carlo and spatial point processes. In Proc. Seminaire Européen de Statistique Toulouse 1996 ‘Stochastic Geometry: Theory and Applications’, ed. Kendall, W. S.. CRC/Chapman and Hall, New York, pp. 141–172.Google Scholar
[10] Penrose, M. D. (1995). Generalized two-sample U-statistics and a two-species reaction-diffusion model. Stoch. Proc. Appl. 55, 57–64.CrossRefGoogle Scholar
[11] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.Google Scholar
[12] Saunders, R. and Funk, G. M. (1977). Poisson limits for a clustering model of Strauss. J. Appl. Prob. 14, 776–784.Google Scholar
[13] Saunders, R., Kryscio, R. J. and Funk, G. M. (1982). Poisson limits for a hard-core clustering model. Stoch. Proc. Appl. 12, 97–106.Google Scholar
[14] Silverman, B. and Brown, T. (1978). Short distances, flat triangles and Poisson limits. J. Appl. Prob. 15, 815–825.Google Scholar
[15] Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and its Applications, 2nd edn, John Wiley, Chichester, UK.Google Scholar
[16] Strauss, D. (1986). On a general class of models for interaction. SIAM Rev. 28, 513–527.Google Scholar
[17] Widom, B. and Rowlinson, J. S. (1970). New model for the study of liquid–vapor phase transitions. J. Chem. Phys. 52, 1670–1684.Google Scholar