  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.
