We study statistical inference for the optimal transport (OT) map (also known as the Brenier map) from a known absolutely continuous reference distribution onto an unknown finitely discrete target distribution. We derive limit distributions for the $L^p$-error with arbitrary $p \in [1,\infty)$ and for linear functionals of the empirical OT map, together with their moment convergence. The former has a non-Gaussian limit, whose explicit density is derived, while the latter attains asymptotic normality. For both cases, we also establish consistency of the nonparametric bootstrap. The derivation of our limit theorems relies on new stability estimates of functionals of the OT map with respect to the dual potential vector, which may be of independent interest. We also discuss applications of our limit theorems to the construction of confidence sets for the OT map and inference for a maximum tail correlation.

翻译：我们研究从已知绝对连续参考分布到未知有限离散目标分布的最优传输（OT）映射（又称Brenier映射）的统计推断问题。针对任意$p\in[1,\infty)$，我们推导了经验OT映射$L^p$误差的极限分布及其线性泛函的极限分布，同时建立了相应的矩收敛性。前者具有非高斯极限分布，我们给出了其显式密度表达式，而后者则满足渐近正态性。对于这两种情形，我们还证明了非参数自助法的一致性。上述极限定理的推导依赖于OT映射泛函关于对偶势向量的新稳定性估计，该估计可能具有独立的研究价值。我们进一步讨论了这些极限定理在构造OT映射置信集以及最大尾部相关性推断中的应用。