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A Compensator Characterization of Planar Point Processes

  • B. Gail IvanoffEmail author
Part of the Fields Institute Communications book series (FIC, volume 76)

Abstract

Martingale techniques play a fundamental role in the analysis of point processes on \([0,\infty )\). In particular, the compensator of a point process uniquely determines and is determined by its distribution, and an explicit formula involving conditional interarrival distributions is well-known. In two dimensions there are many possible definitions of a point process compensator and we focus here on the one that has been the most useful in practice: the so-called *-compensator. Although existence of the *-compensator is well understood, in general it does not determine the law of the point process and it must be calculated on a case-by-case basis. However, it will be proven that when the point process satisfies a certain property of conditional independence (usually denoted by (F4)), the *-compensator determines the law of the point process and an explicit regenerative formula can be given. The basic building block of the planar model is the single line process (a point process with incomparable jump points). Its law can be characterized by a class of avoidance probabilities that are the two-dimensional counterpart of the survival function on \([0,\infty )\). Conditional avoidance probabilities then play the same role in the construction of the *-compensator as conditional survival probabilities do for compensators in one dimension.

Notes

Acknowledgements

Research supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

References

  1. 1.
    Aletti, G., Capasso, V.: Characterization of spatial Poisson along optional increasing paths – a problem of dimension’s reduction. Stat. Probab. Lett. 43, 343–347 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Brown, T.: A martingale approach to the Poisson convergence of simple point processes. Ann. Probab. 6, 615–628 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Brown, T., Ivanoff, B.G., Weber, N.C.: Poisson convergence in two dimensions with applications to row and column exchangeable arrays. Stoch. Proc. Appl. 23, 307–318 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes, vol. II, 2nd edn. Springer, New York (2008)Google Scholar
  5. 5.
    Dozzi, M.: On the decomposition and integration of two-parameter stochastic processes. In: Colloque ENST-CNET. Lecture Notes in Mathematics, vol. 863, pp. 162–171. Springer, Berlin/Heidelberg (1981)Google Scholar
  6. 6.
    Gushchin, A.A.: On the general theory of random fields on the plane. Russ. Math. Surv. 37(6), 55–80 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Ivanoff, B.G., Merzbach, E.: Set-Indexed Martingales. Chapman & Hall/CRC, Boca Raton (2000)zbMATHGoogle Scholar
  8. 8.
    Ivanoff, B.G., Merzbach, E.: Set-indexed Markov processes. Can. Math. Soc. Conf. Proc. 26, 217–232 (2000)MathSciNetGoogle Scholar
  9. 9.
    Ivanoff, B.G., Merzbach, E.: What is a multi-parameter renewal process? Stochastics 78, 411–441 (2006)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Ivanoff, B.G., Merzbach, E., Plante, M.: A compensator characterization of point processes on topological lattices. Electron. J. Prob. 12, 47–74 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Jacod, J.: Multivariate point processes: Predictable projection, Radon-Nikodym derivatives, representation of martingales. Z. Wahr. 31, 235–253 (1975)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Kallenberg, O.: Foundations of Modern Probability. Springer, New York (1997)zbMATHGoogle Scholar
  13. 13.
    Karr, A.F.: Point Processes and Their Statistical Inference, 2nd edn. Marcel Dekker, New York (1991)zbMATHGoogle Scholar
  14. 14.
    Last, G.: Predictable projections for point process filtrations. Probab. Theory Relat. Fields 99, 361–388 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Mazziotto, G., Merzbach, E.: Point processes indexed by directed sets. Stoch. Proc. Appl. 30, 105–119 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Merzbach, E.: Point processes in the plane. Acta Appl. Math. 12, 79–101 (1988)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematics & StatisticsUniversity of OttawaOttawaCanada

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