Limits of compound and thinned point processes

Let r= Y _ be a point process on some space S and let B, l1, B2, ... be identically distributed non-negative rando variables which are mutually independent and independent of r. We can then form the compound point process = flj5 which is a random measure on S. The purpose of this paper is to study the limiting behaviour of ? as B -+ 0. In the particular case when B takes the values 1 and 0 with probabilities p and i-p respectively, becomes a p-thinning of 1 and our theorems contain some classical results by R6nyi and others on the thinnings of a fixed process, as well as a characterization by Mecke of the class of subordinated Poisson processes. COMPOUND AND THINNED POINT PROCESSES; INFINITELY DIVISIBLE RANDOM MEASURES; SUBORDINATED POISSON PROCESSES; CONVERGENCE IN DISTRIBUTION; REGULARITY AND DIFFUSENESS