Void probabilities and factorial moment measures of generalized Matérn hard-core point processes

Several generalizations of the classical Matérn type I and II hard-core point processes have been proposed in the literature during the last decades; for such processes, explicit results for the first- and second-order characteristics are available. We define here general thinning rules, both deterministic and probabilistic, of an inhomogeneous marked Poisson point process, and we provide explicit expressions for the factorial moment measures of any order of the thinned process and of its marginal process. As a byproduct, general expressions for the intensity measure and for the pair correlation function are given, recovering known results in the literature as special cases. In particular, we provide a general expression for the void probability function of the thinned process, from which we deduce upper and lower bounds. Possible applications in the study of the mean volume density of particular birth-and-growth models in materials science are discussed.