Estimation of Stationary Optimal Transport Plans
This work addresses the challenge of analyzing long-run behavior in stationary processes for researchers in probability theory and optimal transport, representing an incremental extension of iid results to the stationary case.
The paper tackles the problem of estimating optimal transport plans for stationary stochastic processes by introducing estimators for optimal joinings and their cost, establishing consistency under mild conditions and finite-sample error rates under stronger mixing assumptions.
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.