We prove that Kendall’s Rank correlation matrix converges to the Marčenko Pastur law, under the assumption that observations are i.i.d random vectors $X_1, \ldots , X_n$ with components that are independent and absolutely continuous with respect to the Lebesgue measure. This is the first result on the empirical spectral distribution of a multivariate $U$-statistic.
Afonso S. Bandeira. Asad Lodhia. Philippe Rigollet. "Marčenko-Pastur law for Kendall’s tau." Electron. Commun. Probab. 22 1 - 7, 2017. https://doi.org/10.1214/17-ECP59