Markov properties of cluster

We show that a Poisson cluster point process is a nearest-neighbour Markov point process 2] if the clusters have uniformly bounded diameter. It is typically not a nite-range Markov point process in the sense of Ripley and Kelly 12]. Furthermore, when the parent Poisson process is replaced by a Markov or nearest-neighbour Markov point process, the resulting cluster process is also nearest-neighbour Markov, provided all clusters are nonempty. In particular, the nearest-neighbour Markov property is preserved when points of the process are independently randomly translated, but not when they are randomly thinned.