On Using the First Difference in Stein - Chen Method

This paper investigates an alternative way of using the Stein-Chen method in Poisson approximations. There are three principal bounds stated in terms of reduced Palm probabilities for general point processes. The rst two are for the accuracy of Poisson random variable approximation to the distribution of the number of points in a point process with respect to the total variation metric and Wasserstein metric, and the third is for bounding the errors of Poisson process approximation to the distribution of a point process on a general compact space with respect to a Wasserstein metric. The bounds are frequently sharper than the previous results using the Stein-Chen method when the expected number of points is large.