## The Annals of Probability

### Ergodicity of Poisson products and applications

Tom Meyerovitch

#### Abstract

In this paper we study the Poisson process over a $\sigma$-finite measure-space equipped with a measure preserving transformation or a group of measure preserving transformations. For a measure-preserving transformation $T$ acting on a $\sigma$-finite measure-space $X$, the Poisson suspension of $T$ is the associated probability preserving transformation $T_{*}$ which acts on realization of the Poisson process over $X$. We prove ergodicity of the Poisson-product $T\times T_{*}$ under the assumption that $T$ is ergodic and conservative. We then show, assuming ergodicity of $T\times T_{*}$, that it is impossible to deterministically perform natural equivariant operations: thinning, allocation or matching. In contrast, there are well-known results in the literature demonstrating the existence of isometry equivariant thinning, matching and allocation of homogenous Poisson processes on $\mathbb{R}^{d}$. We also prove ergodicity of the “first return of left-most transformation” associated with a measure preserving transformation on $\mathbb{R}_{+}$, and discuss ergodicity of the Poisson-product of measure preserving group actions, and related spectral properties.

#### Article information

Source
Ann. Probab., Volume 41, Number 5 (2013), 3181-3200.

Dates
First available in Project Euclid: 12 September 2013

https://projecteuclid.org/euclid.aop/1378991836

Digital Object Identifier
doi:10.1214/12-AOP824

Mathematical Reviews number (MathSciNet)
MR3127879

Zentralblatt MATH identifier
1279.60061

Subjects
Primary: 60G55: Point processes 37A05: Measure-preserving transformations

#### Citation

Meyerovitch, Tom. Ergodicity of Poisson products and applications. Ann. Probab. 41 (2013), no. 5, 3181--3200. doi:10.1214/12-AOP824. https://projecteuclid.org/euclid.aop/1378991836

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