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.
翻译:我们研究固定式随机过程的最佳运输方法,在有限的空间中取用数值。为了反映基本过程的固定性,我们限制对固定式联结(又称联结)的注意。由此产生的最佳合并问题捕捉了利益进程长期平均行为的差异。我们引入了最佳合并和最佳合并成本的估测器,并在温和条件下确定了估计测算器的一致性。此外,在更强的混合假设下,我们为扩大已知最佳合并结果的预期最佳联结成本设定了限定式误差率。最后,我们将一致性和比率分析扩展至最佳合并问题的一个英特质化版本。