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Deterministic thinning of finite Poisson processes

Authors: Omer Angel, Alexander E. Holroyd and Terry Soo
Journal: Proc. Amer. Math. Soc. 139 (2011), 707-720
MSC (2010): Primary 60G55
Published electronically: August 19, 2010
MathSciNet review: 2736350
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Abstract: Let $ \Pi$ and $ \Gamma$ be homogeneous Poisson point processes on a fixed set of finite volume. We prove a necessary and sufficient condition on the two intensities for the existence of a coupling of $ \Pi$ and $ \Gamma$ such that $ \Gamma$ is a deterministic function of $ \Pi$, and all points of $ \Gamma$ are points of $ \Pi$. The condition exhibits a surprising lack of monotonicity. However, in the limit of large intensities, the coupling exists if and only if the expected number of points is at least one greater in $ \Pi$ than in $ \Gamma$.

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Additional Information

Omer Angel
Affiliation: Department of Mathematics, University of British Columbia, 121–1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada

Alexander E. Holroyd
Affiliation: Microsoft Research, 1 Microsoft Way, Redmond, Washington 98052

Terry Soo
Affiliation: Department of Mathematics, University of British Columbia, 121–1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada

Received by editor(s): December 2, 2009
Received by editor(s) in revised form: December 4, 2009, and April 12, 2010
Published electronically: August 19, 2010
Additional Notes: Funded in part by Microsoft Research (AEH) and NSERC (all authors)
Communicated by: Edward C. Waymire
Article copyright: © Copyright 2010 American Mathematical Society

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