Asymptotics for Strassen's Optimal Transport Problem
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In this paper, we consider Strassen's version of optimal transport problem. That is, we minimize the excess-cost probability (i.e., the probability that the cost is larger than a given value) over all couplings of two distributions. We derive large deviation, moderate deviation, and central limit theorems for this problem. Our approach is based on Strassen's dual formulation of the optimal transport 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 optimal transport problem, a nested optimal transport formula for Strassen's optimal transport problem is derived. In this nested formula, the cost function of the outer optimal transport subproblem is set to the optimal transport functional (i.e., the mapping from a pair of distributions to the optimal transport cost for this pair of distributions) of the inner optimal transport subproblem. Based on this nested formula, we carefully design asymptotically optimal solutions to Strassen's optimal transport problem and its dual formulation. Finally, we connect Strassen's optimal transport problem to the empirical optimal transport problem, which hence provides an application for our results.
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