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 $L^p$-estimation error with arbitrary $p \in [1,\infty)$ and for linear functionals of the empirical OT map. The former has a non-Gaussian limit, 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 could be of independent interest.

翻译：我们研究了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映射（又称Brenier映射）的统计推断。我们推导了任意 $p \in [1,\infty)$ 下 $L^p$ 估计误差的极限分布，以及经验最优传输映射线性泛函的极限分布。前者具有非高斯极限，后者则满足渐近正态性。在两种情形下，我们还建立了非参数自助法的相合性。极限定理的推导依赖于最优传输映射泛函关于对偶势向量的新稳定性估计，该结果本身可能具有独立研究价值。