Poisson thickening |
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Authors: | Ori Gurel-Gurevich Ron Peled |
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Institution: | 1. University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada 2. School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel
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Abstract: | Let × be a Poisson point process of intensity λ on the real line. A thickening of it is a (deterministic) measurable function f such that X∪f(X) is a Poisson point process of intensity λ′ where λ′ > λ. An equivariant thickening is a thickening which commutes with all shifts of the line. We show that a thickening exists but an equivariant thickening does not. We prove similar results for thickenings which commute only with integer shifts and in the discrete and multi-dimensional settings. This answers 3 questions of Holroyd, Lyons and Soo. We briefly consider also a much more general setup in which we ask for the existence of a deterministic coupling satisfying a relation between two probability measures. We present a conjectured sufficient condition for the existence of such couplings. |
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