  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.
