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September 2013 Ergodicity of Poisson products and applications
Tom Meyerovitch
Ann. Probab. 41(5): 3181-3200 (September 2013). DOI: 10.1214/12-AOP824

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.

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Tom Meyerovitch. "Ergodicity of Poisson products and applications." Ann. Probab. 41 (5) 3181 - 3200, September 2013. https://doi.org/10.1214/12-AOP824

Information

Published: September 2013
First available in Project Euclid: 12 September 2013

zbMATH: 1279.60061
MathSciNet: MR3127879
Digital Object Identifier: 10.1214/12-AOP824

Subjects:
Primary: 37A05 , 60G55

Keywords: conservative transformations , equivariant allocation , equivariant thinning , infinite measure preserving transformations , Poisson suspension

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 5 • September 2013
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