Palm theory, random measures and Stein couplings

We establish a general Berry--Esseen type bound which gives optimal bounds in many situations under suitable moment assumptions. By combining the general bound with Palm theory, we deduce a new error bound for assessing the accuracy of normal approximation to statistics arising from random measures, including stochastic geometry. We illustrate the use of the bound in three examples: completely random measures, excursion random measure of a locally dependent random process and the total edge length of Ginibre-Voronoi tessellations. We also consider the general result in the context of Stein couplings and discuss various cases of Stein couplings, including local dependence, exchangeable pairs and the size-bias coupling, and occupancy problems.