×

zbMATH — the first resource for mathematics

Deterministic thinning of finite Poisson processes. (English) Zbl 1208.60047
Summary: Let \( \Pi\) and \( \Gamma\) be homogeneous Poisson point processes on a fixed set of finite volume. We prove a necessary and sufficient condition on the two intensities for the existence of a coupling of \( \Pi\) and \( \Gamma\) such that \( \Gamma\) is a deterministic function of \( \Pi\), and all points of \( \Gamma\) are points of \( \Pi\). The condition exhibits a surprising lack of monotonicity. However, in the limit of large intensities, the coupling exists if and only if the expected number of points is at least one greater in \( \Pi\) than in \( \Gamma\).

MSC:
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Karen Ball, Poisson thinning by monotone factors, Electron. Comm. Probab. 10 (2005), 60 – 69. · Zbl 1110.60050 · doi:10.1214/ECP.v10-1134 · doi.org
[2] M. Brand. Using Your Head Is Permitted. http://www.brand.site.co.il/riddles, March 2009.
[3] S. Evans. A zero-one law for linear transformations of Lévy noise. In Proceedings of the Conference on Algebraic Methods in Statistics and Probability, Contemporary Mathematics Series, 2010. To appear.
[4] O. Gurel-Gurevich and R. Peled. Poisson thickening. Preprint, arXiv:0911.5377.
[5] Alexander E. Holroyd and Yuval Peres, Extra heads and invariant allocations, Ann. Probab. 33 (2005), no. 1, 31 – 52. · Zbl 1097.60032 · doi:10.1214/009117904000000603 · doi.org
[6] J. F. C. Kingman, Poisson processes, Oxford Studies in Probability, vol. 3, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. · Zbl 0771.60001
[7] R. Lyons, A. Holroyd, and T. Soo. Poisson splitting by factors. Preprint, arXiv:0908.3409. · Zbl 1277.60087
[8] R.-D. Reiss, A course on point processes, Springer Series in Statistics, Springer-Verlag, New York, 1993. · Zbl 0771.60037
[9] Frank Spitzer, A combinatorial lemma and its application to probability theory, Trans. Amer. Math. Soc. 82 (1956), 323 – 339. · Zbl 0071.13003
[10] S. M. Srivastava, A course on Borel sets, Graduate Texts in Mathematics, vol. 180, Springer-Verlag, New York, 1998. · Zbl 0903.28001
[11] V. Strassen, The existence of probability measures with given marginals, Ann. Math. Statist. 36 (1965), 423 – 439. · Zbl 0135.18701 · doi:10.1214/aoms/1177700153 · doi.org
[12] Hermann Thorisson, Coupling, stationarity, and regeneration, Probability and its Applications (New York), Springer-Verlag, New York, 2000. · Zbl 0949.60007
[13] Peter Winkler, Mathematical puzzles: a connoisseur’s collection, A K Peters, Ltd., Natick, MA, 2004. · Zbl 1094.00003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.