  We derive limit distributions for certain empirical regularized optimal
transport distances between probability distributions supported on a finite
metric space and show consistency of the (naive) bootstrap. In particular, we
prove that the empirical regularized transport plan itself asymptotically
follows a Gaussian law. The theory includes the Boltzmann-Shannon entropy
regularization and hence a limit law for the widely applied Sinkhorn
divergence. Our approach is based on an application of the implicit function
theorem to necessary and sufficient optimality conditions for the regularized
transport problem. The asymptotic results are investigated in Monte Carlo
simulations. We further discuss computational and statistical applications,
e.g. confidence bands for colocalization analysis of protein interaction
networks based on regularized optimal transport.
