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 $\frac{1}{2}pK_n$ converges almost surely to the Mar\v{c}enko--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 $\frac{1}{2}tr\Sigma K_n$. The theoretical findings are validated through comprehensive simulation studies.

翻译：多元Kendall-$\tau$统计量（记为$K_n$）在稳健统计分析中具有重要作用。本文建立了$K_n$的经验谱分布（ESD）的极限性质。我们证明了$\frac{1}{2}pK_n$的ESD几乎必然收敛于方差参数为$\frac{1}{2}$的Mar\v{c}enko--Pastur律，这与样本协方差矩阵的经典结果类似。利用Stieltjes变换技术，我们将这些结果推广至独立成分模型，推导出一个刻画$\frac{1}{2}tr\Sigma K_n$极限谱分布的定点方程。理论结果通过全面的模拟研究得到了验证。