  For testing two random vectors for independence, we consider testing whether
the distance of one vector from a center point is independent from the distance
of the other vector from a center point by a univariate test. In this paper we
provide conditions under which it is enough to have a consistent univariate
test of independence on the distances to guarantee that the power to detect
dependence between the random vectors increases to one, as the sample size
increases. These conditions turn out to be minimal. If the univariate test is
distribution-free, the multivariate test will also be distribution-free. If we
consider multiple center points and aggregate the center-specific univariate
tests, the power may be further improved, and the resulting multivariate test
may be distribution-free for specific aggregation methods (if the univariate
test is distribution-free). We show that several multivariate tests recently
proposed in the literature can be viewed as instances of this general approach.
