  We treat the problem of testing independence between m continuous variables
when m can be larger than the available sample size n. We consider three types
of test statistics that are constructed as sums or sums of squares of pairwise
rank correlations. In the asymptotic regime where both m and n tend to
infinity, a martingale central limit theorem is applied to show that the null
distributions of these statistics converge to Gaussian limits, which are valid
with no specific distributional or moment assumptions on the data. Using the
framework of U-statistics, our result covers a variety of rank correlations
including Kendall's tau and a dominating term of Spearman's rank correlation
coefficient (rho), but also degenerate U-statistics such as Hoeffding's $D$, or
the $\tau^*$ of Bergsma and Dassios (2014). As in the classical theory for
U-statistics, the test statistics need to be scaled differently when the rank
correlations used to construct them are degenerate U-statistics. The power of
the considered tests is explored in rate-optimality theory under Gaussian
equicorrelation alternatives as well as in numerical experiments for specific
cases of more general alternatives.
