Distance Estimates for Poisson Process Approximations of Dependent Thinnings

Abstract

It is well known, that under certain conditions, gradual thinning of a point process on $R^d_+$, accompanied by a contraction of space to compensate for the thinning, leads in the weak limit to a Cox process. In this article, we apply discretization and a result based on Stein's method to give estimates of the Barbour-Brown distance $d_2$ between the distribution of a thinned point process and an approximating Poisson process, and evaluate the estimates in concrete examples. We work in terms of two, somewhat different, thinning models. The main model is based on the usual thinning notion of deleting points independently according to probabilities supplied by a random field. In Section 4, however, we use an alternative thinning model, which can be more straightforward to apply if the thinning is determined by point interactions.