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Poisson splitting by factors
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Holroyd, Alexander E., Lyons, Russell and Soo, Terry (2011) Poisson splitting by factors. Annals of Probability, Vol.39 (No.5). pp. 1938-1982. doi:10.1214/11-AOP651 ISSN 0091-1798.
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Official URL: http://dx.doi.org/10.1214/11-AOP651
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.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||
Journal or Publication Title: | Annals of Probability | ||||
Publisher: | Institute of Mathematical Statistics | ||||
ISSN: | 0091-1798 | ||||
Official Date: | September 2011 | ||||
Dates: |
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Volume: | Vol.39 | ||||
Number: | No.5 | ||||
Page Range: | pp. 1938-1982 | ||||
DOI: | 10.1214/11-AOP651 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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