Thinning of Cluster Processes: Convergence of Sums of Thinned Point Processes

This is a study of the convergence in distribution of sums of dependent point processes that are becoming uniformly sparse due to a thinning operation. Under this operation, each point process is randomly deleted or retained depending on its structure, and when a process is retained, each of its points is deleted or retained depending on its location and the structure of the process. Such a sum can be interpreted as a thinned cluster process: the residual of a cluster process after its cluster origins and single points have been thinned. Our main result gives a necessary and sufficient condition for the sums to converge and states that their limit must be a Cox process (a Poisson process with a random intensity measure). This result has some parallels to the classical result on the convergence of sums of independent point processes to a Poisson process, and it contains Kallenberg's result on the convergence of a thinned point process to a Cox process. Corollaries are presented for the cases in which the p...