Point shift characterization of Palm measures on Abelian groups

Abstract

Our first aim in this paper is to characterize Palm measures of stationary point processes through point stationarity. This generalizes earlier results from the Euclidean case to the case of an Abelian group. While a stationary point process looks statistically the same from each site, a point stationary point process looks statistically the same from each of its points. Even in the Euclidean case our proof will simplify some of the earlier arguments. A new technical result of some independent interest is the existence of a complete countable family of matchings. Using a change of measure we will generalize our results to discrete random measures. In the Euclidean case we will finally treat general random measures by means of a suitable approximation.