Electronic Communications in Probability

Poisson Thinning by Monotone Factors

Karen Ball

Full-text: Open access


Let $X$ and $Y$ be Poisson point processes on the real numbers with rates $l_1$ and $l_2$ respectively. We show that if $l_1 > l_2$, then there exists a deterministic map $f$ such that $f(X)$ and $Y$ have the same distribution, the joint distribution of $(X, f(X))$ is translation-invariant, and which is monotone in the sense that for all intervals $I$, $f(X)(I) \leq X(I)$, almost surely.

Article information

Electron. Commun. Probab., Volume 10 (2005), paper no. 7, 60-69.

Accepted: 16 April 2005
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

This work is licensed under aCreative Commons Attribution 3.0 License.


Ball, Karen. Poisson Thinning by Monotone Factors. Electron. Commun. Probab. 10 (2005), paper no. 7, 60--69. doi:10.1214/ECP.v10-1134. https://projecteuclid.org/euclid.ecp/1465058072

Export citation


  • Ball, K. Monotone factors of i.i.d. processes, to appear in the Israel J. Math.
  • Ferrari, P.A., Landim, C., Thorisson, H. Poisson trees, succession lines and coalescing random walks, Ann. I. H. Poincaré-PR 40 (2004), 141-152.
  • Holroyd, A., Peres, Y. Trees and matchings from point processes, Elect. Comm. in Probab. 8 (2003), 17-27.
  • Keane, M., Smorodinsky, M. A class of finitary codes, Israel J. Math. 26 (1977) nos. 3-4, 352-371.
  • Keane, M., Smorodinsky, M. Bernoulli schemes of the same entropy are finitarily isomorphic, Ann.Math. 109 (1979), 397-406.
  • Reiss, R.-D. A course on point processes, Springer-Verlag, New York, 1993.
  • Strassen, V. The existence of probability measures with given marginals, Ann. Math. Statist. 36 (1965), 423-439.
  • Timár, Á. Tree and grid factors for general point processes, Elect. Comm. in Probab. 9 (2004), 53-59.