The Annals of Probability

Ergodicity of Poisson products and applications

Tom Meyerovitch

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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

Permanent link to this document
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

Keywords
Poisson suspension equivariant thinning equivariant allocation infinite measure preserving transformations conservative 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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References

  • [1] Aaronson, J. (1997). An Introduction to Infinite Ergodic Theory. Mathematical Surveys and Monographs 50. Amer. Math. Soc., Providence, RI.
  • [2] Aaronson, J. and Nadkarni, M. (1987). $L_{\infty}$ eigenvalues and $L_{2}$ spectra of nonsingular transformations. Proc. Lond. Math. Soc. (3) 55 538–570.
  • [3] Adler, R. L. and Weiss, B. (1973). The ergodic infinite measure preserving transformation of Boole. Israel J. Math. 16 263–278.
  • [4] Ball, K. (2005). Poisson thinning by monotone factors. Electron. Commun. Probab. 10 60–69 (electronic).
  • [5] Chatterjee, S., Peled, R., Peres, Y. and Romik, D. (2010). Gravitational allocation to Poisson points. Ann. of Math. (2) 172 617–671.
  • [6] Evans, S. N. (2010). A zero-one law for linear transformations of Lévy noise. In Algebraic Methods in Statistics and Probability II. Contemp. Math. 516 189–197. Amer. Math. Soc., Providence, RI.
  • [7] Gurel-Gurevich, O. and Peled, R. (2013). Poisson thickening. Israel J. Math. To appear. Available at arXiv:0911.5377.
  • [8] Hahn, P. (1979). Reconstruction of a factor from measures on Takesaki’s unitary equivalence relation. J. Funct. Anal. 31 263–271.
  • [9] Hoffman, C., Holroyd, A. E. and Peres, Y. (2006). A stable marriage of Poisson and Lebesgue. Ann. Probab. 34 1241–1272.
  • [10] Holroyd, A. E. (2011). Geometric properties of Poisson matchings. Probab. Theory Related Fields 150 511–527.
  • [11] Holroyd, A. E., Lyons, R. and Soo, T. (2011). Poisson splitting by factors. Ann. Probab. 39 1938–1982.
  • [12] Holroyd, A. E., Pemantle, R., Peres, Y. and Schramm, O. (2009). Poisson matching. Ann. Inst. Henri Poincaré Probab. Stat. 45 266–287.
  • [13] Kingman, J. F. C. (1993). Poisson Processes. Oxford Studies in Probability 3. The Clarendon Press Oxford Univ. Press, New York.
  • [14] Kingman, J. F. C. (2006). Poisson processes revisited. Probab. Math. Statist. 26 77–95.
  • [15] Krikun, M. (2007). Connected allocation to Poisson points in $\mathbb{R}^{2}$. Electron. Commun. Probab. 12 140–145.
  • [16] Nadkarni, M. G. (2011). Spectral Theory of Dynamical Systems. Texts and Readings in Mathematics 15. Hindustan Book Agency, New Delhi.
  • [17] Roy, E. (2009). Poisson suspensions and infinite ergodic theory. Ergodic Theory Dynam. Systems 29 667–683.
  • [18] Roy, E. (2010). Poisson–Pinsker factor and infinite measure preserving group actions. Proc. Amer. Math. Soc. 138 2087–2094.
  • [19] Schmidt, K. (1982). Spectra of ergodic group actions. Israel J. Math. 41 151–153.