arXiv:2604.23928v2 Announce Type: replace Abstract: In this paper, we establish upper and lower bounds on the convergence rate of empirical measures of point processes under the Wasserstein distance. To this end, we first introduce a new metric on the space of finite counting measures. Unlike many existing metrics in the literature, our proposed metric incorporates the spatial locations of unmatched points. Drawing on both this new metric and existing metrics from the literature, we identify a set of common structural properties and develop a general framework for Wasserstein distances between point-process distributions based on metrics satisfying these properties. We then use this framework to investigate the convergence rates of empirical measures, providing quantitative guarantees for approximating the distribution of the underlying point process, and further establish corresponding concentration inequalities. These theoretical results may provide ingredients for constructing oracle goodness-of-fit tests for a specified point-process distribution. We also apply our theory to the approximation of Campbell measures and discuss how the resulting bounds can help assess Wasserstein-based learning of generative point-process models.