  For probability measures on countable spaces we derive distributional limits
for empirical entropic optimal transport quantities. More precisely, we show
that the empirical optimal transport plan weakly converges to a centered
Gaussian process and that the empirical entropic optimal transport value is
asymptotically normal. The results are valid for a large class of cost
functions and generalize distributional limits for empirical entropic optimal
transport quantities on finite spaces. Our proofs are based on a sensitivity
analysis with respect to norms induced by suitable function classes, which
arise from novel quantitative bounds for primal and dual optimizers, that are
related to the exponential penalty term in the dual formulation. The
distributional limits then follow from the functional delta method together
with weak convergence of the empirical process in that respective norm, for
which we provide sharp conditions on the underlying measures. As a byproduct of
our proof technique, consistency of the bootstrap for statistical applications
is shown.
