Poisson thickening

Ori Gurel-Gurevich*, Ron Peled

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Pages (from-to)215-234
Number of pages20
JournalIsrael Journal of Mathematics
Volume196
Issue number1
DOIs
StatePublished - Aug 2013

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