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Poisson splitting by factors
Holroyd, Alexander E. Lyons, Russell Soo, Terry
Holroyd, Alexander E.
Lyons, Russell
Soo, Terry
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Abstract
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.
Description
This is the published version, also available here: http://dx.doi.org/10.1214/11-AOP651.
Date
2011-11-02
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Institute of Mathematical Statistics
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Keywords
Poisson process, stochastic domination, factor map, thinning
Citation
Holroyd, Alexander E.; Lyons, Russell; Soo, Terry. Poisson splitting by factors. Ann. Probab. 39 (2011), no. 5, 1938--1982. http://dx.doi.org/10.1214/11-AOP651.