The empirical copula process, a fundamental tool for copula inference, is studied in the high dimensional regime where the dimension is allowed to grow to infinity exponentially in the sample size. Under natural, weak smoothness assumptions on the underlying copula, it is shown that Stute's representation is valid in the following sense: all low-dimensional margins of fixed dimension of the empirical copula process can be approximated by a functional of the low-dimensional margins of the standard empirical process, with the almost sure error term being uniform in the margins. The result has numerous potential applications, and is exemplary applied to the problem of testing pairwise stochastic independence in high dimensions, leading to various extensions of recent results in the literature: for certain test statistics based on pairwise association measures, type-I error control is obtained for models beyond mutual independence. Moreover, bootstrap-based critical values are shown to yield strong control of the familywise error rate for a large class of data generating processes.

翻译：经验联结函数过程是联结函数推断的基本工具，本文研究其在允许维数随样本量呈指数级增长的高维框架下的性质。在底层联结函数满足自然弱光滑性假设的条件下，证明Stute表示在以下意义下成立：经验联结函数过程所有固定维数的低维边沿可由标准经验过程的低维边沿函数逼近，且几乎必然误差项在边沿上具有一致性。该结果具有众多潜在应用，并作为示范应用于高维成对随机独立性检验问题，从而推广了文献中的近期结论：对于基于成对关联测度的特定检验统计量，可在超越互独立性的模型中获得第一类错误控制。此外，基于自助法的临界值被证明能对一大类数据生成过程实现家族误差率的强控制。