Limit Distributions and Sensitivity Analysis for Entropic Optimal Transport on Countable Spaces

For probability measures supported on countable spaces we derive limit distributions for empirical entropic optimal transport quantities. In particular, we prove that the corresponding plan converges weakly to a centered Gaussian process. Furthermore, its optimal value is shown to be asymptotically normal. The results are valid for a large class of ground cost functions and generalize recently obtained limit laws for empirical entropic optimal transport quantities on finite spaces. Our proofs are based on a sensitivity analysis with respect to a weighted $\ell^1$-norm relying on the dual formulation of entropic optimal transport as well as necessary and sufficient optimality conditions for the entropic transport plan. This can be used to derive weak convergence of the empirical entropic optimal transport plan and value that results in weighted Borisov-Dudley-Durst conditions on the underlying probability measures. The weights are linked to an exponential penalty term for dual entropic optimal transport and the underlying ground cost function under consideration. Finally, statistical applications, such as bootstrap, are discussed.