Poisson Thickening
Abstract
Let X be a Poisson point process of intensity lambda on the real line. A thickening of it is a (deterministic) measurable function f such that the union of X and f(X) is a Poisson point process of intensity lambda' where lambda'>lambda. 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.
Cite
@article{arxiv.0911.5377,
title = {Poisson Thickening},
author = {Ori Gurel-Gurevich and Ron Peled},
journal= {arXiv preprint arXiv:0911.5377},
year = {2017}
}
Comments
Added conjecture about when a deterministic coupling satisfying a relation exists. Made some minor revisions. To appear in Israel Journal of Mathematics. 16 pages