Testing mutual independence in high dimensions with sums of squares of rank correlations

We treat the problem of testing mutual independence between m continuous observations when the available sample size n is comparable to m. As test statistics, we consider sums of squared rank correlations between pairs of random variables. Specific examples we study are sums of squares formed from Kendall's tau and from Spearman's rho. In the asymptotic regime where m/n converges to a positive constant, the null distribution of these statistics is shown to converge to a normal limit. The proofs are based on results for sums of squares of rank-based U-statistics.