Extremality of poisson and sample processes

Improving and extending a characterization of Poisson processes by Rényi, we present several mild conditions ensuring a point process to be of Poisson or sample type, and we give related conditions for convergence in distribution towards such processes. The basic property underlying our results is the extremality of pure Poisson and sample processes within the classes of mixed ones, the latter being the only symmetrically distributed simple point processes. As applications, we extend some results of Rényi and Kendall on interchangeable events.