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In this article, we study tests of independence for data with arbitrarydistributions in the non-serial case, i.e., for independent and identicallydistributed random vectors, as well as in the serial case, i.e., for timeseries. These tests are derived from copula-based covariances and theirmultivariate extensions using M\"obius transforms.  We find the asymptotic distributions of these statistics under the nullhypothesis of independence or randomness, as well as under contiguousalternatives. This enables us to find out locally most powerful test statisticsfor some alternatives, whatever the margins. Numerical experiments areperformed for Wald's type combinations of these statistics to assess the finitesample performance.

Tests of independence and randomness for arbitrary data using copula-based covariances