Limits of compound and thinned point processes

Abstract

Let η =Σ_jδ_τj be a point process on some space S and let β,β₁,β₂, … be identically distributed non-negative random variables which are mutually independent and independent of η. We can then form the compound point process ξ = Σ_jβ_jδ_τj which is a random measure on S. The purpose of this paper is to study the limiting behaviour of ξ as . In the particular case when β takes the values 1 and 0 with probabilities p and 1 –p respectively, ξ becomes a p-thinning of η and our theorems contain some classical results by Rényi and others on the thinnings of a fixed process, as well as a characterization by Mecke of the class of subordinated Poisson processes.