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Spatial Jump Processes and Perfect Simulation

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Morphology of Condensed Matter

Part of the book series: Lecture Notes in Physics ((LNP,volume 600))

Abstract

Spatial birth-and-death processes, spatial birth-and-catastrophe processes, and more general types of spatial jump processes are studied in detail. Particularly, various kinds of coupling constructions are considered, leading to some known and some new perfect simulation procedures for the equilibrium distributions of different types of spatial jump processes. These equilibrium distributions include many classical Gibbs point process models and a new class of models for spatial point processes introduced in the text.

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References

  1. Asmussen, S. (1987): Applied Probability and Queues. (Wiley, Chichester)

    MATH  Google Scholar 

  2. Baddeley, A. and van Lieshout, M. N. M. (1995): ‘Area-interaction Point Processes’. Ann. Inst. Statist. Math. 46, pp. 601–619.

    Article  Google Scholar 

  3. Berthelsen, K. K. and Møller, J. (2001): ‘Perfect simulation and inference for spatial point processes’. Technical report, R-01-2009, Department of Mathematical Sciences, Aalborg University. Submitted.

    Google Scholar 

  4. Berthelsen, K. K. and Møller, J. (2001): ‘A primer on perfect simulation for spatial point processes’. Technical report, R-01-2026, Department of Mathematical Sciences, Aalborg University. Submitted.

    Google Scholar 

  5. Daley, D. J. and Vere-Jones, D. (1988): An Introduction to Point Processes. (Springer, Berlin)

    MATH  Google Scholar 

  6. Döge, G. (2000): ‘Grand canonical simulations of hard-disk systems’. In: Statistical Physics and Spatial Statistics, ed. by K. R. Mecke and D. Stoyan, volume 554 of Lecture Notes in Physics (Springer, Berlin), pp. 373–393.

    Chapter  Google Scholar 

  7. Feller, W. (1971): An Introduction to Probability Theory and Its Applications, volume 2 (John Wiley and Sons Inc., New York), 2nd edition.

    MATH  Google Scholar 

  8. Fernández, R., Ferrari, P. A. and Garcia, N. L. (1999): ‘Perfect simulation for interacting point processes, loss networks and Ising models’. Unpublished.

    Google Scholar 

  9. Georgii, H.-O. (2000): ‘Phase transitions and percolation in Gibbsian particle models’. In: Statistical Physics and Spatial Statistics, ed. by K. R. Mecke and D. Stoyan, volume 554 of Lecture Notes in Physics (Springer, Berlin), pp. 267–294.

    Chapter  Google Scholar 

  10. Geyer, C. J. (1999): ‘Likelihood inference for spatial processes’. In: Stochastic Geometry, Likelihood and Computation, ed. by O. E. Barndorff-Nielsen, W. Kendall and M. van Lieshout (Chapman & Hall), pp. 79–140.

    Google Scholar 

  11. Häggström, O., van Lieshout, M.-C. N. M. and Møller, J. (1999): ‘Characterization results and Markovchain Monte Carlo algorithms including exact simulation for some spatial point processes’. Bernoulli 5, pp. 641–658.

    Article  MATH  MathSciNet  Google Scholar 

  12. Kallenberg, O. (1984): ‘An informal guide to the theory of conditioning in point processes’. Int. Statist. Rev. 52, pp. 151–164.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Kelly, F. P. and Ripley, B. D. (1976): ‘A note on Strauss’s model for clustering’. Biometrika 63, pp. 357–360.

    Article  MATH  MathSciNet  Google Scholar 

  14. Kendall, W. S. (1998): ‘Perfect Simulation for the Area-Interaction point process’. In: Probability Towards 2000, ed. by L. Accardi and C. Heyde (Springer, New York), pp. 218–234.

    Google Scholar 

  15. Kendall, W. S. and Møller, J. (2000): ‘Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes’. Adv. Appl. Prob. 32, pp. 844–865.

    Article  MATH  Google Scholar 

  16. Lieshout, M. N. M. van (2000): Markov point processes and their applications. Imperial College Press, London.

    Book  MATH  Google Scholar 

  17. Löwen, H. (2000): ‘Fun with hard spheres’. In: Statistical Physics and Spatial Statistics, ed. by K. R. Mecke and D. Stoyan, volume 554 of Lecture Notes in Physics (Springer, Berlin), pp. 295–331.

    Chapter  Google Scholar 

  18. Mase, S., Møller, J., Stoyan, D., Waagepetersen, R. and Döge, G. (2001): ‘Packing densities and simulated tempering for hard core Gibbs point processes’, Ann. Inst. Statis. Math. 53, pp. 661–680.

    Article  MATH  Google Scholar 

  19. Mecke, K. (2000): ‘Additivity, convexity and beyond: applications of Minkowski functionals in statistical physics’. In: Statistical Physics and Spatial Statistics, ed. by K. R. Mecke and D. Stoyan, volume 554 of Lecture Notes in Physics (Springer, Berlin), pp. 111–184.

    Chapter  Google Scholar 

  20. Meyn, S. P. and Tweedie, R. L. (1994): Markov Chains and Stochastic Stability. (Springer-Verlag, London)

    Google Scholar 

  21. Møller, J. (1989): ‘On the rate of convergence of spatial birth-and-death processes’. Ann. Inst. Statist. Math. 41, pp. 565–581.

    Article  MATH  MathSciNet  Google Scholar 

  22. Møller, J. (2001): ‘A review of perfect simulation in stochastic geometry’. In: Selected Proceedings of the Symposium on Inference for Stochastic Processes, ed. by I. V. Basawa, C. C. Heyde and R. L. Taylor, volume 37 of IMS Lecture Notes & Monographs Series, pp. 333–355.

    Google Scholar 

  23. Møller, J. and Schladitz, K. (1999): ‘Extensions of Fill’s algorithm for perfect simulation’. J. Roy. Statist. Soc. Ser. B 61, pp. 1–15.

    Article  Google Scholar 

  24. Norris, J. R. (1997): Markov Chains. (Cambridge University Press, New York)

    MATH  Google Scholar 

  25. Preston, C. (1977): ‘Spatial birth-and-death processes’. Bull. Int. Statist. Inst. 46, pp. 371–391.

    MathSciNet  Google Scholar 

  26. Propp, J. G. and Wilson, D. B. (1996): ‘Exact Sampling with Coupled MarkovChains and Applications to Statistical Mechanics’. Random Structures and Algorithms 9, pp. 223–252.

    Article  MATH  MathSciNet  Google Scholar 

  27. Reuter, G. E. H. and Ledermann, W. (1953): ‘On the differential equations for the transition probabilities of Markovprocesses with enumerably many states’. Proc. Camb. Phil. Soc. 49, pp. 247–262.

    Article  MATH  MathSciNet  Google Scholar 

  28. Ripley, B. D. (1977): ‘Modelling spatial patterns (with discussion)’. J. Roy. Statist. Soc. Ser. B 39, pp. 172–212.

    MathSciNet  Google Scholar 

  29. Ruelle, D. (1969): Statistical Mechanis: Rigorous Results. (W. A. Benjamin, Reading, Massachusetts)

    MATH  Google Scholar 

  30. Strauss, D. J. (1975): ‘A model for clustering’. Biometrika 62, pp. 467–475.

    Article  MATH  MathSciNet  Google Scholar 

  31. Thönnes, E. (1999): ‘Perfect simulation of some point processes for the impatient user’. Adv. Appl. Prob. 31, pp. 69–87.

    Article  MATH  Google Scholar 

  32. Thönnes, E. (2000): ‘A primer on perfect simulation’. In: Statistical Physics and Spatial Statistics, ed. by K.R. Mecke and D. Stoyan, volume 554 of Lecture Notes in Physics (Springer, Berlin), pp. 349–378.

    Chapter  Google Scholar 

  33. Widom, B. and Rowlinson, J. S. (1970): ‘A new model for the study of liquid-vapor phase transitions’. J. Chem. Phys. 52, pp. 1670–1684.

    Article  ADS  Google Scholar 

  34. Wilson, D. B. (2000): ‘How to Couple from the Past Using a Read-Once Source of Randomness’. Random Structures and Algorithms 16, pp. 85–113.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220, Aalborg, Denmark

    Kasper K. Berthelsen & Jesper Møller

Authors
  1. Kasper K. Berthelsen
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  2. Jesper Møller
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Editor information

Editors and Affiliations

  1. Max-Planck-Institut für Metallforschung, Heisenbergstrasse 1, 70569, Stuttgart, Germany

    Klaus Mecke

  2. Fakultät für Mathematik und Informatik Institut für Stochastik, TU Bergakademie Freiberg, 09596, Freiberg, Germany

    Dietrich Stoyan

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Berthelsen, K.K., Møller, J. (2002). Spatial Jump Processes and Perfect Simulation. In: Mecke, K., Stoyan, D. (eds) Morphology of Condensed Matter. Lecture Notes in Physics, vol 600. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45782-8_16

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  • DOI: https://doi.org/10.1007/3-540-45782-8_16

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-44203-5

  • Online ISBN: 978-3-540-45782-4

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