Asymptotic Laws and Methods in Stochastics pp 87-109 | Cite as

# A Compensator Characterization of Planar Point Processes

## 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.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.Brown, T.: A martingale approach to the Poisson convergence of simple point processes. Ann. Probab.
**6**, 615–628 (1978)zbMATHMathSciNetCrossRefGoogle Scholar - 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.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.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.Gushchin, A.A.: On the general theory of random fields on the plane. Russ. Math. Surv.
**37**(6), 55–80 (1982)zbMATHMathSciNetCrossRefGoogle Scholar - 7.Ivanoff, B.G., Merzbach, E.: Set-Indexed Martingales. Chapman & Hall/CRC, Boca Raton (2000)zbMATHGoogle Scholar
- 8.Ivanoff, B.G., Merzbach, E.: Set-indexed Markov processes. Can. Math. Soc. Conf. Proc.
**26**, 217–232 (2000)MathSciNetGoogle Scholar - 9.Ivanoff, B.G., Merzbach, E.: What is a multi-parameter renewal process? Stochastics
**78**, 411–441 (2006)zbMATHMathSciNetGoogle Scholar - 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.Jacod, J.: Multivariate point processes: Predictable projection, Radon-Nikodym derivatives, representation of martingales. Z. Wahr.
**31**, 235–253 (1975)zbMATHMathSciNetCrossRefGoogle Scholar - 12.Kallenberg, O.: Foundations of Modern Probability. Springer, New York (1997)zbMATHGoogle Scholar
- 13.Karr, A.F.: Point Processes and Their Statistical Inference, 2nd edn. Marcel Dekker, New York (1991)zbMATHGoogle Scholar
- 14.Last, G.: Predictable projections for point process filtrations. Probab. Theory Relat. Fields
**99**, 361–388 (1994)zbMATHMathSciNetCrossRefGoogle Scholar - 15.Mazziotto, G., Merzbach, E.: Point processes indexed by directed sets. Stoch. Proc. Appl.
**30**, 105–119 (1988)zbMATHMathSciNetCrossRefGoogle Scholar - 16.Merzbach, E.: Point processes in the plane. Acta Appl. Math.
**12**, 79–101 (1988)zbMATHMathSciNetGoogle Scholar