Poisson convergence for point processes on the plane

Abstract

A compensator is defined for a point process in two dimensions. It is shown that a Poisson process is characterized by a continuous deterministic compensator. Sufficient conditions are given for convergence in distribution of a sequence of two-dimensional point processes in the Skorokhod topology to a Poisson process when the corresponding sequence of compensators converges pointwise in probability to a continuous deterministic function.1980 Mathematics subject classification (Amer. Math. Soc): primary 60 G 55; secondary 60 F 17, 60 G 48.