We consider the copula mapping, which maps a joint cumulative distribution function to the corresponding copula. Its Hadamard differentiablity was shown in van der Vaart and Wellner (1996), Fermanian et al. (2004) and (under less strict assumptions) in B\"ucher and Volgushev (2013). This differentiability result has proved to be a powerful tool to show weak convergence of empirical copula processes in various settings using the functional delta method. We state a generalization of the Hadamard differentiability results that simplifies the derivations of asymptotic expansions and weak convergence of empirical copula processes in the presence of covariates. The usefulness of this result is illustrated on several applications which include a multidimensional functional linear model, where the copula of the error vector describes the dependency between the components of the vector of observations, given the functional covariate.

翻译：本文研究copula映射，该映射将联合累积分布函数映射至相应的copula。其哈达玛可微性已在van der Vaart与Wellner（1996）、Fermanian等（2004）以及（在较宽松假设下）Bücher与Volgushev（2013）的研究中得到证明。该可微性结果被证明是一种强有力的工具，可通过泛函delta方法，在不同场景下展示经验copula过程的弱收敛性。我们提出了一种广义的哈达玛可微性结果，简化了存在协变量时经验copula过程渐近展开式与弱收敛性的推导过程。该结果的实用性通过若干应用得以展示，其中包括一个多维泛函线性模型，在该模型中，误差向量的copula描述了观测向量各分量之间的依赖关系（在给定泛函协变量的条件下）。