We study limit theorems for entropic optimal transport (EOT) maps, dual potentials, and the Sinkhorn divergence. The key technical tool we use is a first and second-order Hadamard differentiability analysis of EOT potentials with respect to the marginal distributions, which may be of independent interest. Given the differentiability results, the functional delta method is used to obtain central limit theorems for empirical EOT potentials and maps. The second-order functional delta method is leveraged to establish the limit distribution of the empirical Sinkhorn divergence under the null. Building on the latter result, we further derive the null limit distribution of the Sinkhorn independence test statistic and characterize the correct order. Since our limit theorems follow from Hadamard differentiability of the relevant maps, as a byproduct, we also obtain bootstrap consistency and asymptotic efficiency of the empirical EOT map, potentials, and Sinkhorn divergence.

翻译：我们研究熵最优传输(EOT)映射、对偶势函数及Sinkhorn散度的极限定理。本文采用的关键技术工具是对EOT势函数关于边际分布的一阶和二阶Hadamard可微性分析，该分析本身可能具有独立研究价值。基于可微性结果，我们利用函数delta方法推导出经验EOT势函数和映射的中心极限定理。借助二阶函数delta方法，我们建立了零假设下经验Sinkhorn散度的极限分布。基于后者结果，进一步导出Sinkhorn独立性检验统计量的零极限分布并刻画其正确阶数。由于相关映射的Hadamard可微性保证了极限定理的有效性，作为副产品，我们还获得了经验EOT映射、势函数及Sinkhorn散度的自举一致性与渐近有效性。