Asymptotics for Strassen's Optimal Transport Problem

In this paper, we consider Strassen's version of optimal transport (OT) problem, which concerns minimizing the excess-cost probability (i.e., the probability that the cost is larger than a given value) over all couplings of two given distributions. We derive large deviation, moderate deviation, and central limit theorems for this problem. Our proof is based on Strassen's dual formulation of the OT problem, Sanov's theorem on the large deviation principle (LDP) of empirical measures, as well as the moderate deviation principle (MDP) and central limit theorems (CLT) of empirical measures. In order to apply the LDP, MDP, and CLT to Strassen's OT problem, nested formulas for Strassen's OT problem are derived. Based on these nested formulas and using a splitting technique, we construct asymptotically optimal solutions to Strassen's OT problem and its dual formulation.