Statistical optimal transport is not governed by one universal central limit theorem. Transport costs can be smooth away from equality and degenerate at equality; potentials may be unique or set-valued; maps require differentiation of convex potentials; barycenters are nonlinear Fréchet means. The inferential regime must be identified before a limit law or bootstrap is chosen.
Transport value as a statistical functional
Definition. For a fixed cost \(c\), write \[ \mathcal T_c(\mu,\nu) =\inf_{\pi\in\Pi(\mu,\nu)}\int c\,\mathrm{d}\pi. \] Its plug-in estimator is \(\mathcal T_c(\widehat\mu_n,\widehat\nu_m)\), or \(\mathcal T_c(\widehat\mu_n,\nu)\) in the one-sample case.
Directional derivatives from duality
Envelope derivative. Let \(\Phi^\star(\mu,\nu)\) be the normalized set of optimal dual pairs. Under compactness and continuity conditions ensuring dual attainment and stable potentials, the directional derivative in signed zero-mass directions \((\alpha,\beta)\) is \[ \mathcal T'_{c,(\mu,\nu)}(\alpha,\beta) = \sup_{(\varphi,\psi)\in\Phi^\star(\mu,\nu)} \left\{\int\varphi\,\mathrm{d}\alpha+\int\psi\,\mathrm{d}\beta\right\}. \] If the normalized dual pair is unique, the derivative is linear.
Proof. Duality writes the value as the supremum of affine maps \[ (\mu,\nu)\longmapsto\int\varphi\,\mathrm{d}\mu+\int\psi\,\mathrm{d}\nu \] over the cost-dependent feasible set. For a perturbation of size \(t\), any optimizer at the base point gives a lower derivative bound. Conversely, choose optimizers at the perturbed measures; compactness produces a limiting optimal pair at the base point, giving the matching upper bound. This is Danskin's envelope argument in the topology in which potential integrals are continuous.
Uniqueness of the transport plan does not by itself guarantee differentiability of the value. What matters for the envelope derivative is uniqueness of the normalized dual potentials in the relevant function space.
Central limit theorem in a regular regime
Two-sample delta method. Suppose the normalized optimal potentials \((\varphi,\psi)\) are unique, square integrable, and the empirical processes admit the required functional convergence. If \(n/(n+m)\to\lambda\in(0,1)\), then \[ \sqrt{\frac{nm}{n+m}} \{\mathcal T_c(\widehat\mu_n,\widehat\nu_m)-\mathcal T_c(\mu,\nu)\} \Rightarrow N(0,\sigma^2), \] where \[ \sigma^2 =(1-\lambda)\operatorname{Var}_\mu\{\varphi(X)\} +\lambda\operatorname{Var}_\nu\{\psi(Y)\}. \]
Proof. Linear differentiability gives \[ \mathcal T_c(\widehat\mu_n,\widehat\nu_m)-\mathcal T_c(\mu,\nu) = (\widehat\mu_n-\mu)\varphi+(\widehat\nu_m-\nu)\psi +o_{\mathbb P}(n^{-1/2}+m^{-1/2}). \] The two empirical averages are independent. The ordinary central limit theorem and the limiting sample-size ratio give the displayed variance.
Nonunique potentials and directional limits
If \(\Phi^\star\) contains several normalized pairs, the derivative is a supremum of Gaussian linear functionals rather than a single linear functional. The functional delta method then yields a generally non-Gaussian limit \[ \sup_{(\varphi,\psi)\in\Phi^\star} \{\mathbb G_\mu\varphi+\mathbb G_\nu\psi\}. \] This occurs naturally for discrete measures, symmetric configurations, and costs with flat optimal faces.
Degeneracy under equality
First-order degeneracy. For \(c(x,y)=d(x,y)^p\), \(\mathcal T_c(\mu,\mu)=0\). The zero potential pair is optimal, and in many smooth settings the first directional derivative along probability perturbations vanishes. The empirical transport cost under the null is then governed by a second-order or nonsmooth limit and does not have the same scaling as under \(\mu\ne\nu\).
One-dimensional \(W_2\) null limit. Suppose \(\mu\) has a positive smooth density \(f\) on a compact interval and quantile \(Q=F^{-1}\). The quantile process satisfies \[ \sqrt n\{F_n^{-1}(u)-Q(u)\} \Rightarrow-\frac{B(u)}{f(Q(u))}, \] where \(B\) is a Brownian bridge. Therefore \[ nW_2^2(\widehat\mu_n,\mu) \Rightarrow \int_0^1\frac{B(u)^2}{f(Q(u))^2}\,\mathrm{d}u. \] The limit is quadratic and non-Gaussian, illustrating why a root-\(n\) Gaussian approximation to \(W_2^2\) fails at the null.
Bootstrap validity
Bootstrap principle. The ordinary empirical bootstrap is typically valid when \(\mathcal T_c\) is fully Hadamard differentiable with a unique linear derivative and the potential class is empirically regular. It may fail when the derivative is only directional, when the null is degenerate, or when the empirical support changes the active optimal face.
Possible remedies include subsampling, an \(m\)-out-of-\(n\) bootstrap, estimating the directional derivative and simulating its Gaussian argument, or using entropic regularization to obtain a smooth functional. Each remedy targets a different source of nonregularity and must be justified for the regime at hand.
Estimating optimal potentials
Empirical dual potentials are \(M\)-estimators over a constrained function class. Consistency follows from compactness after normalization, uniform convergence of the empirical dual objective, and uniqueness of the population potential. Rates require local curvature of the dual criterion and control of the complexity of the potential class. Potentials are statistically important because they are influence functions for regular transport costs.
Estimating transport maps
Plug-in map principle. For quadratic cost, estimate densities or potentials, solve an empirical or regularized Monge--Ampère problem for \(\widehat u\), and set \(\widehat T=\nabla\widehat u\). Under strong convexity and regularity, \[ \|\widehat T-T\|_{L^2(\mu)} \] can be controlled by errors in the estimated potential, its gradient, or the source and target measures in stronger norms than weak convergence alone.
Raw empirical measures in dimension \(d\ge2\) usually yield a discrete plan rather than a globally defined map. A map estimator therefore requires interpolation, density smoothing, semi-discrete transport, or another structural regularization. Stability of costs alone is insufficient: differentiating a convex potential is an ill-posed operation unless additional smoothness is imposed.
Statistical barycenters
Fréchet \(M\)-estimation principle. Let \(\mu_1,\ldots,\mu_N\) be sampled from a law \(\mathsf Q\) on \(\mathcal P_2\), and let \[ \widehat{\bar\mu}_N\in\operatorname*{arg\,min}_\nu \frac1{2N}\sum_{i=1}^NW_2^2(\nu,\mu_i). \] If the population barycenter is unique and second moments are integrable, then \(\widehat{\bar\mu}_N\to\bar\mu\) under standard compactness assumptions. If the Fréchet functional has a nonsingular second variation in a tangent chart, then \[ \sqrt N\,\log_{\bar\mu}(\widehat{\bar\mu}_N) \Rightarrow H^{-1}G, \] where \(G\) is the Gaussian limit of empirical tangent scores and \(H\) is the Hessian of the population Fréchet functional.
Proof mechanism. Uniform laws of large numbers on moment-bounded sets give consistency by the argmin theorem. For the limit law, expand the empirical first-order barycenter equation around \(\bar\mu\): \[ 0=\frac1N\sum_i S(\bar\mu,\mu_i) +H\,\log_{\bar\mu}(\widehat{\bar\mu}_N)+o_{\mathbb P}(N^{-1/2}). \] The first term obeys a central limit theorem in the tangent space, and inversion of \(H\) gives the result. Curvature, nonuniqueness, or singular measures can invalidate this linear chart.
Regularization changes the estimand
Entropic and other regularizations often restore differentiability and improve sample behavior, but they estimate a regularized transport functional. Valid inference must separate stochastic error from regularization bias and specify whether the regularization parameter is fixed or tends to zero. Pages 23 and 24 make this tradeoff explicit.