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Point-map-probabilities of a point process and Mecke's invariant measure equation

Baccelli, F ; Sharif University of Technology | 2017

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  1. Type of Document: Article
  2. DOI: 10.1214/16-AOP1099
  3. Publisher: Institute of Mathematical Statistics , 2017
  4. Abstract:
  5. A compatible point-shift F maps, in a translation invariant way, each point of a stationary point process Φ to some point of Φ. 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 Φ 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 Φ 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 Φ boils down to the Palm probability of Φ. When it is not bijective, there exist cases where the point-map-probability of Φ is singular with respect to its Palm probability. A tightness based criterion for the existence of the pointmap- 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. © Institute of Mathematical Statistics, 2017
  6. Keywords:
  7. Allocation rule ; Dynamical system ; Mass transport principle ; Palm probability ; Point process ; Point-map ; Point-shift ; Stationarity ; Vague topology ; ω-limit set
  8. Source: Annals of Probability ; Volume 45, Issue 3 , 2017 , Pages 1723-1751 ; 00911798 (ISSN)
  9. URL: https://projecteuclid.org/euclid.aop/1494835229