Some limit theorems for clustered occupancy models

Most generalizations of the classical occupancy model involve non-homogeneous shot assignment probabilities, but retain the independence of the individual shot assignments. Hence, these models are associated with non-homogeneous Poisson processes. The present article discusses a generalization in which the shot assignments are not independent, but which result in clustering of the shots. Conditions are given under which this clustered occupancy model converges to a Poisson cluster process. Limiting distributions for the number of empty cells are obtained for various allocation intensities when the total number of shots is deterministic as well as random. In particular, it is shown that when the allocation is sparse, then the limiting distribution of the number of empty cells is compound Poisson.