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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.
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
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