On mixing,compounding, and tail properties of a class of claim number distributions

The mathematical structure underlying a class of discrete claim count distributions is examined in detail. In particular, the mixed Poisson nature of the class is shown to hold fairly generally. Using some ideas involving complete monotonicity, a discussion is provided on the structure of other class members which are well suited for use in aggregate claims analysis. The ideas are then extended to the analysis of the corresponding discrete tail probabilities, which arise in a variety of contexts including the analysis of the stop-loss premium.