COUNT DISTRIBUTIONS, ORDERLINESS AND INVARIANCE OF POISSON CLUSTER PROCESSES.

The probability generating functional (p. g. fl. ) of a non-homogeneous Poisson cluster process is characterized in L. P. Ammann and P. F. Thall via a decomposition of the KLM measure of the process. This p. g. fl. representation is utilized to show that the family D of Poisson cluster processes with a. s. finite clusters is invariant under a class of cluster transformations. Explicit expressions for the finite-dimensional count distributions, product moment measures, and the distribution of clusters are derived in terms of the KLM measure. It is also shown that an element of D has no multiple events if the points of each cluster are a. s. distinct.