Central Limit Theorems for Optimal Transport Functionals

We establish central limit theorems for empirical plug-in estimators of scalar functionals of quadratic optimal transport maps and potentials, including Wasserstein costs and several risk measures constructed from transport-based multivariate quantiles. We study the one-sample problem with a known reference distribution and i.i.d. observations from the target distribution. A natural approach is to combine a limit theorem for the empirical transport map with the functional delta method. We show, however, that in dimensions at least two, empirical transport-map errors admit no nonzero weak limit in $L^2$ under any normalizing scale. Instead, we linearize the functional directly in the target distribution, obtaining an influence-function expansion with a remainder controlled by the transport-map error. The influence function is characterized by a linear elliptic partial differential equation. Under regularity conditions, we use this expansion to prove central limit theorems in low dimensions and establish validity of the empirical bootstrap. We also derive semiparametric efficiency bounds and show that the plug-in estimators attain them under the same conditions.