Point-stationarity in \(d\) dimensions and Palm theory
The author extends to \(d>1\) dimensions the concept of point-stationarity, which formalizes the intuitive idea of a simple point process for which the behavior relative to a given point of the process is independent of the point selected as origin. Note that point-stationarity is different from stationarity. For \(d=1\), the Poisson process with constant intensity is stationary since if the origin is moved from 0 to any other fixed site \(t\), then we still have a Poisson process with the same intensity. But if a point is added at 0, then we obtain a non-stationary process since if the origin is moved from the point which we placed at 0 to any other fixed site \(t\), then there need not be a point there. This new process is point-stationary since, seen from the point which we placed at 0, the intervals between points are i.i.d. exponential. After defining point-stationarity, this concept is characterized in several ways and the characterizations then are used to extend to \(d\) dimensions a particular approach to Palm theory.
scientific article; zbMATH DE number 1396205