Distance estimates for dependent superpositions of point processes

In this article, superpositions of possibly dependent point processes on a general space X are considered. Using Stein's method for Poisson process approximation, an estimate is given for the Wasserstein distance d2 between the distribution of such a superposition and an appropriate Poisson process distribution. This estimate is compared to a modern version of Grigelionis' theorem, and to results of Banys [Lecture Notes in Statistics, vol. 2, Springer, New York, 1980, pp. 26-37], Arratia et al. [Ann. Probab. 17 (1989) 9-25] and Barbour et al. [Poisson Approximation, Oxford University Press, Oxford, 1992]. Furthermore, an application to a spatial birth-death model is presented.