Poisson Thickening

TL;DR

This paper investigates the existence of deterministic functions that can increase the intensity of Poisson processes while preserving their properties, showing that such functions exist generally but equivariant ones do not, answering open questions.

Contribution

It proves the existence of non-equivariant thickenings of Poisson processes and the non-existence of equivariant ones, addressing open problems in the field.

Findings

Existence of non-equivariant thickenings of Poisson processes.

Non-existence of equivariant thickenings under shift invariance.

Extension of results to discrete and multi-dimensional settings.

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.