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We derive distributional limits for empirical transport distances betweenprobability measures supported on countable sets. Our approach is based onsensitivity analysis of optimal values of infinite dimensional mathematicalprograms and a delta method for non-linear derivatives. A careful calibrationof the norm on the space of probability measures is needed in order to combinedifferentiability and weak convergence of the underlying empirical process.Based on this we provide a sufficient and necessary condition for theunderlying distribution on the countable metric space for such a distributionallimit to hold. We give an explicit form of the limiting distribution forultra-metric spaces. Finally, we apply our findings to optimal transport basedinference in large scale problems. An application to nanoscale microscopy isgiven.

Empirical optimal transport on countable metric spaces: Distributional limits and statistical applications