The Annals of Probability

Poisson splitting by factors

Alexander E. Holroyd, Russell Lyons, and Terry Soo

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Given a homogeneous Poisson process on ℝd with intensity λ, we prove that it is possible to partition the points into two sets, as a deterministic function of the process, and in an isometry-equivariant way, so that each set of points forms a homogeneous Poisson process, with any given pair of intensities summing to λ. In particular, this answers a question of Ball [Electron. Commun. Probab. 10 (2005) 60–69], who proved that in d = 1, the Poisson points may be similarly partitioned (via a translation-equivariant function) so that one set forms a Poisson process of lower intensity, and asked whether the same is possible for all d. We do not know whether it is possible similarly to add points (again chosen as a deterministic function of a Poisson process) to obtain a Poisson process of higher intensity, but we prove that this is not possible under an additional finitariness condition.

Article information

Ann. Probab., Volume 39, Number 5 (2011), 1938-1982.

First available in Project Euclid: 18 October 2011

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Zentralblatt MATH identifier

Primary: 60G55: Point processes 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10]

Poisson process stochastic domination factor map thinning


Holroyd, Alexander E.; Lyons, Russell; Soo, Terry. Poisson splitting by factors. Ann. Probab. 39 (2011), no. 5, 1938--1982. doi:10.1214/11-AOP651.

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