Electronic Journal of Probability

Point-shift foliation of a point process

Francois Baccelli and Mir-Omid Haji-Mirsadeghi

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A point-shift $F$ maps each point of a point process $\Phi $ to some point of $\Phi $. For all translation invariant point-shifts $F$, the $F$-foliation of $\Phi $ is a partition of the support of $\Phi $ which is the discrete analogue of the stable manifold of $F$ on $\Phi $. It is first shown that foliations lead to a classification of the behavior of point-shifts on point processes. Both qualitative and quantitative properties of foliations are then established. It is shown that for all point-shifts $F$, there exists a point-shift $F_\bot $, the orbits of which are the $F$-foils of $\Phi $, and which is measure-preserving. The foils are not always stationary point processes. Nevertheless, they admit relative intensities with respect to one another.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 19, 25 pp.

Received: 30 October 2016
Accepted: 4 November 2017
First available in Project Euclid: 23 February 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37C85: Dynamics of group actions other than Z and R, and foliations [See mainly 22Fxx, and also 57R30, 57Sxx] 60G10: Stationary processes 60G55: Point processes 60G57: Random measures

point process stationarity Palm probability point-shift point-map allocation rule mass transport principle dynamical system stable manifold

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Baccelli, Francois; Haji-Mirsadeghi, Mir-Omid. Point-shift foliation of a point process. Electron. J. Probab. 23 (2018), paper no. 19, 25 pp. doi:10.1214/17-EJP123. https://projecteuclid.org/euclid.ejp/1519354948

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