  We consider the behavior of spatial point processes when subjected to a class
of linear transformations indexed by a variable T. It was shown in Ellis [Adv.
in Appl. Probab. 18 (1986) 646-659] that, under mild assumptions, the
transformed processes behave approximately like Poisson processes for large T.
In this article, under very similar assumptions, explicit upper bounds are
given for the d_2-distance between the corresponding point process
distributions. A number of related results, and applications to kernel density
estimation and long range dependence testing are also presented. The main
results are proved by applying a generalized Stein-Chen method to discretized
versions of the point processes.
