## The Annals of Probability

- Ann. Probab.
- Volume 39, Number 5 (2011), 1938-1982.

### Poisson splitting by factors

Alexander E. Holroyd, Russell Lyons, and Terry Soo

#### Abstract

Given a homogeneous Poisson process on ℝ^{d} with intensity *λ*, we prove that it is possible to partition the points into two sets, as a deterministic function of the process, and in an isometry-equivariant way, so that each set of points forms a homogeneous Poisson process, with any given pair of intensities summing to *λ*. In particular, this answers a question of Ball [*Electron. Commun. Probab.* **10** (2005) 60–69], who proved that in *d* = 1, the Poisson points may be similarly partitioned (via a translation-equivariant function) so that *one* set forms a Poisson process of lower intensity, and asked whether the same is possible for all *d*. We do not know whether it is possible similarly to *add* points (again chosen as a deterministic function of a Poisson process) to obtain a Poisson process of higher intensity, but we prove that this is not possible under an additional finitariness condition.

#### Article information

**Source**

Ann. Probab., Volume 39, Number 5 (2011), 1938-1982.

**Dates**

First available in Project Euclid: 18 October 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1318940786

**Digital Object Identifier**

doi:10.1214/11-AOP651

**Mathematical Reviews number (MathSciNet)**

MR2884878

**Zentralblatt MATH identifier**

1277.60087

**Subjects**

Primary: 60G55: Point processes 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10]

**Keywords**

Poisson process stochastic domination factor map thinning

#### Citation

Holroyd, Alexander E.; Lyons, Russell; Soo, Terry. Poisson splitting by factors. Ann. Probab. 39 (2011), no. 5, 1938--1982. doi:10.1214/11-AOP651. https://projecteuclid.org/euclid.aop/1318940786