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Testing for pairwise independence for the case where the number of variablesmay be of the same size or even larger than the sample size has receivedincreasing attention in the recent years. We contribute to this branch of theliterature by considering tests that allow to detect higher-order dependencies.The proposed methods are based on connecting the problem to copulas and makinguse of the Moebius transformation of the empirical copula process; an approachthat has already been used successfully for the case where the number ofvariables is fixed. Based on a martingale central limit theorem, it is shownthat respective test statistics converge to the standard normal distribution,allowing for straightforward definition of critical values. The results areillustrated by a Monte Carlo simulation study.

Testing for independence in high dimensions based on empirical copulas