POINT-MAP-PROBABILITIES OF A POINT PROCESS AND MECKE'S INVARIANT MEASURE EQUATION
成果类型：
Article
署名作者：
Baccelli, Francois; Haji-Mirsadeghij, Mir-Omid
署名单位：
University of Texas System; University of Texas Austin; Sharif University of Technology
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/16-AOP1099
发表日期：
2017
页码：
1723-1751
关键词：
摘要：
A compatible point-shift F maps, in a translation invariant way, each point of a stationary point process (1) 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 non-bijective 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 1 boils down to the Palm probability of Phi. When it is not bijective, there exist cases where the point-map-probability of CD 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.