  We study limit theorems for entropic optimal transport (EOT) maps, dual
potentials, and the Sinkhorn divergence. The key technical tool we use is a
first and second-order Hadamard differentiability analysis of EOT potentials
with respect to the marginal distributions, which may be of independent
interest. Given the differentiability results, the functional delta method is
used to obtain central limit theorems for empirical EOT potentials and maps.
The second-order functional delta method is leveraged to establish the limit
distribution of the empirical Sinkhorn divergence under the null. Building on
the latter result, we further derive the null limit distribution of the
Sinkhorn independence test statistic and characterize the correct order. Since
our limit theorems follow from Hadamard differentiability of the relevant maps,
as a byproduct, we also obtain bootstrap consistency and asymptotic efficiency
of the empirical EOT map, potentials, and Sinkhorn divergence.
