  We derive distributional limits for empirical transport distances between
probability measures supported on countable sets. Our approach is based on
sensitivity analysis of optimal values of infinite dimensional mathematical
programs and a delta method for non-linear derivatives. A careful calibration
of the norm on the space of probability measures is needed in order to combine
differentiability and weak convergence of the underlying empirical process.
Based on this we provide a sufficient and necessary condition for the
underlying distribution on the countable metric space for such a distributional
limit to hold. We give an explicit form of the limiting distribution for
ultra-metric spaces. Finally, we apply our findings to optimal transport based
inference in large scale problems. An application to nanoscale microscopy is
given.
