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Point shift characterization of Palm measures on abelian groups. (English) Zbl 1128.60004
Summary: Let $${\mathbf N}$$ denote the space of locally finite simple counting measures on an Abelian topological group $$G$$ that is assumed to be a locally compact, second countable Hausdorff space. A probability measure on $${\mathbf N}$$ is a canonical model of a random point process on $$G$$. Our first aim is to characterize Palm measures of stationary measures on $${\mathbf N}$$ through point stationarity. This generalizes the main result in [the authors, Ann. Probab. 33, No. 5, 1698–1715 (2005; Zbl 1111.60029)] from the Euclidean case to the case of an Abelian group. While under a stationary measure a point process looks statistically the same from each site in $$G$$, under a point stationary point measure it looks statistically the same from each of its points. Even in case $$G=\mathbb R^d$$ our proof simplifies some of the arguments in (loc. cit.). A new technical result of some independent interest is the existence of a complete countable family of matchings. Using a change of measure we generalize our results to discrete random measures. In case $$G=\mathbb R^d$$ we finally treat general random measures by means of a suitable approximation.

##### MSC:
 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60G57 Random measures
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