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September 2011 Poisson splitting by factors
Alexander E. Holroyd, Russell Lyons, Terry Soo
Ann. Probab. 39(5): 1938-1982 (September 2011). DOI: 10.1214/11-AOP651


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.


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Alexander E. Holroyd. Russell Lyons. Terry Soo. "Poisson splitting by factors." Ann. Probab. 39 (5) 1938 - 1982, September 2011.


Published: September 2011
First available in Project Euclid: 18 October 2011

zbMATH: 1277.60087
MathSciNet: MR2884878
Digital Object Identifier: 10.1214/11-AOP651

Primary: 37A50 , 60G55

Keywords: factor map , Poisson process , stochastic domination , thinning

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.39 • No. 5 • September 2011
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