  We study optimal transport for stationary stochastic processes taking values
in finite spaces. In order to reflect the stationarity of the underlying
processes, we restrict attention to stationary couplings, also known as
joinings. The resulting optimal joining problem captures differences in the
long run average behavior of the processes of interest. We introduce estimators
of both optimal joinings and the optimal joining cost, and we establish
consistency of the estimators under mild conditions. Furthermore, under
stronger mixing assumptions we establish finite-sample error rates for the
estimated optimal joining cost that extend the best known results in the iid
case. Finally, we extend the consistency and rate analysis to an
entropy-penalized version of the optimal joining problem.
