In this talk we try to connect five keywords in statistics/probability — rank correlations, (degenerate) U-statistics, combinatorial (non-)CLT, optimal transport theory, and Le Cam’s contiguity lemmas  – through one theme, nonparametric independence testing. The corresponding results show the existence of consistent and statistically efficient distribution-free tests of two null hypotheses, mutual independence and independence of two random vectors, both for the first time. In technical terms, we give (1) the first Cramer-type moderate deviation theorem for degenerate U-statistics, (2) a new type of combinatorial non-central limit theorem for double- and multiple-indexed permutation statistics, and (3) a nontrivial use of Le Cam’s third lemma with elements of non-normal limits.