Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T03:03:45.443Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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