  This paper investigates the problem of testing independence of two random
vectors of general dimensions. For this, we give for the first time a
distribution-free consistent test. Our approach combines distance covariance
with the center-outward ranks and signs developed in Hallin (2017). In
technical terms, the proposed test is consistent and distribution-free in the
family of multivariate distributions with nonvanishing (Lebesgue) probability
densities. Exploiting the (degenerate) U-statistic structure of the distance
covariance and the combinatorial nature of Hallin's center-outward ranks and
signs, we are able to derive the limiting null distribution of our test
statistic. The resulting asymptotic approximation is accurate already for
moderate sample sizes and makes the test implementable without requiring
permutation. The limiting distribution is derived via a more general result
that gives a new type of combinatorial non-central limit theorem for double-
and multiple-indexed permutation statistics.
