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

Abstract

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 processes being summed are either independent or satisfy a weak law of large numbers before they are thinned. In addition, sufficient conditions are given for sums of randomly selected processes, with no single point deletions, to converge to infinitely divisible point processes or to mixtures of these processes.