GROMOV-HAUSDORFF CONVERGENCE OF DISCRETE TRANSPORTATION METRICS

This paper continues the investigation of "Wasserstein-like" transportation distances for probability measures on discrete sets. We prove that the discrete transportation metrics W-N on the d-dimensional discrete torus T-N(d) with mesh size (1)(N) converge, when N -> infinity, to the standard 2-Wasserstein distance W-2 on the continuous torus in the sense of Gromov-Hausdorff. This is the first convergence result for the recently developed discrete transportation metrics W. The result shows the compatibility between these metrics and the well-established 2-Wasserstein metric.