Limiting Spectral Distribution of High-dimensional Multivariate Kendall-$\tau$

The multivariate Kendall-$\tau$ statistic, denoted by $K_n$, plays a significant role in robust statistical analysis. This paper establishes the limiting properties of the empirical spectral distribution (ESD) of $K_n$. We demonstrate that the ESD of $\frac{1}{2}p{K}_n$ converges almost surely to the Marčenko–Pastur law with variance parameter $\frac{1}{2}$, analogous to the classical result for sample covariance matrices. Using Stieltjes transform techniques, we extend these results to the independent component model, deriving a fixed-point equation that characterizes the limiting spectral distribution of $\frac{1}{2}tr\mathrm{\Sigma }{K}_n$. The theoretical findings are validated through comprehensive simulation studies.