  Deheuvels [J. Multivariate Anal. 11 (1981) 102--113] and Genest and
R\'{e}millard [Test 13 (2004) 335--369] have shown that powerful rank tests of
multivariate independence can be based on combinations of asymptotically
independent Cram\'{e}r--von Mises statistics derived from a M\"{o}bius
decomposition of the empirical copula process. A result on the large-sample
behavior of this process under contiguous sequences of alternatives is used
here to give a representation of the limiting distribution of such test
statistics and to compute their relative local asymptotic efficiency. Local
power curves and asymptotic relative efficiencies are compared under familiar
classes of copula alternatives.
