×

zbMATH — the first resource for mathematics

Ergodic Poisson splittings. (English) Zbl 07226360
Summary: In this paper, we study splittings of a Poisson point process which are equivariant under a conservative transformation. We show that, if the Cartesian powers of this transformation are all ergodic, the only ergodic splitting is the obvious one, that is, a collection of independent Poisson processes. We apply this result to the case of a marked Poisson process: under the same hypothesis, the marks are necessarily independent of the point process and i.i.d. Under additional assumptions on the transformation, a further application is derived, giving a full description of the structure of a random measure invariant under the action of the transformation.
MSC:
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G57 Random measures
37A50 Dynamical systems and their relations with probability theory and stochastic processes
37A40 Nonsingular (and infinite-measure preserving) transformations
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] Adams, T., Friedman, N. and Silva, C. E. (1997). Rank-one weak mixing for nonsingular transformations. Israel J. Math. 102 269-281. · Zbl 0896.58039
[2] Ball, K. (2005). Poisson thinning by monotone factors. Electron. Commun. Probab. 10 60-69. · Zbl 1110.60050
[3] Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Vol. I: Elementary Theory and Methods, 2nd ed. Probability and Its Applications (New York). Springer, New York. · Zbl 1026.60061
[4] Danilenko, A. I. (2018). Infinite measure preserving transformations with Radon MSJ. Israel J. Math. 228 21-51. · Zbl 1403.28015
[5] Furstenberg, H. (1967). Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory 1 1-49. · Zbl 0146.28502
[6] Glasner, E. (2003). Ergodic Theory via Joinings. Mathematical Surveys and Monographs 101. Amer. Math. Soc., Providence, RI.
[7] Glasner, S. (1983). Quasifactors in ergodic theory. Israel J. Math. 45 198-208. · Zbl 0528.46047
[8] Holroyd, A. E., Lyons, R. and Soo, T. (2011). Poisson splitting by factors. Ann. Probab. 39 1938-1982. · Zbl 1277.60087
[9] Janvresse, É., Roy, E. and de la Rue, T. (2017). Poisson suspensions and Sushis. Ann. Sci. Éc. Norm. Supér. (4) 50 1301-1334. · Zbl 1382.37006
[10] Janvresse, É., Roy, E. and de la Rue, T. (2019). Nearly finite Chacon Transformation. Ann. H. Lebesgue 2 369-414. · Zbl 1435.37013
[11] Kakutani, S. and Parry, W. (1963). Infinite measure preserving transformations with “mixing”. Bull. Amer. Math. Soc. 69 752-756. · Zbl 0126.31801
[12] Kallenberg, O. (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling 77. Springer, Cham. · Zbl 1376.60003
[13] Meyerovitch, T. (2011). Quasi-factors and relative entropy for infinite-measure-preserving transformations. Israel J. Math. 185 43-60. · Zbl 1348.37011
[14] Meyerovitch, T. (2013). Ergodicity of Poisson products and applications. Ann. Probab. 41 3181-3200. · Zbl 1279.60061
[15] Roy, E. · Zbl 1146.60031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.