  In this article, we study tests of independence for data with arbitrary
distributions in the non-serial case, i.e., for independent and identically
distributed random vectors, as well as in the serial case, i.e., for time
series. These tests are derived from copula-based covariances and their
multivariate extensions using M\"obius transforms.
  We find the asymptotic distributions of these statistics under the null
hypothesis of independence or randomness, as well as under contiguous
alternatives. This enables us to find out locally most powerful test statistics
for some alternatives, whatever the margins. Numerical experiments are
performed for Wald's type combinations of these statistics to assess the finite
sample performance.
