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May 2017 Point-map-probabilities of a point process and Mecke’s invariant measure equation
François Baccelli, Mir-Omid Haji-Mirsadeghi
Ann. Probab. 45(3): 1723-1751 (May 2017). DOI: 10.1214/16-AOP1099

Abstract

A compatible point-shift $F$ maps, in a translation invariant way, each point of a stationary point process $\Phi$ to some point of $\Phi$. It is fully determined by its associated point-map, $f$, which gives the image of the origin by $F$. It was proved by J. Mecke that if $F$ is bijective, then the Palm probability of $\Phi$ is left invariant by the translation of $-f$. The initial question motivating this paper is the following generalization of this invariance result: in the nonbijective case, what probability measures on the set of counting measures are left invariant by the translation of $-f$? The point-map-probabilities of $\Phi$ are defined from the action of the semigroup of point-map translations on the space of Palm probabilities, and more precisely from the compactification of the orbits of this semigroup action. If the point-map-probability exists, is uniquely defined and if it satisfies certain continuity properties, it then provides a solution to this invariant measure problem. Point-map-probabilities are objects of independent interest. They are shown to be a strict generalization of Palm probabilities: when $F$ is bijective, the point-map-probability of $\Phi$ boils down to the Palm probability of $\Phi$. When it is not bijective, there exist cases where the point-map-probability of $\Phi$ is singular with respect to its Palm probability. A tightness based criterion for the existence of the point-map-probabilities of a stationary point process is given. An interpretation of the point-map-probability as the conditional law of the point process given that the origin has $F$-pre-images of all orders is also provided. The results are illustrated by a few examples.

Citation

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François Baccelli. Mir-Omid Haji-Mirsadeghi. "Point-map-probabilities of a point process and Mecke’s invariant measure equation." Ann. Probab. 45 (3) 1723 - 1751, May 2017. https://doi.org/10.1214/16-AOP1099

Information

Received: 1 August 2014; Revised: 1 January 2016; Published: May 2017
First available in Project Euclid: 15 May 2017

zbMATH: 1375.60091
MathSciNet: MR3650413
Digital Object Identifier: 10.1214/16-AOP1099

Subjects:
Primary: 60G10, 60G55, 60G57
Secondary: 60F17, 60G30

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 3 • May 2017
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