Empirical Spectral Distribution of K_n

• The empirical spectral distribution (ESD) of K_n is the study of eigenvalues of the rank‑based Kendall‑τ matrix, showing convergence to the normalized Marčenko–Pastur law.
• The analysis employs the Stieltjes transform to derive fixed-point equations that accurately characterize the limiting spectral behavior in a high-dimensional regime.
• Extensive simulation studies confirm that the ESD of K_n remains invariant under scale transformations and extends to independent component models, ensuring robust multivariate inference.

The empirical spectral distribution (ESD) of $K_n$ refers to the statistical behavior of the eigenvalues of the high-dimensional multivariate Kendall-$\tau$ matrix, a rank-based statistic widely used in robust multivariate analysis. Recent work rigorously establishes the limiting spectral distribution of this matrix in the high-dimensional regime ($p, n \to\infty$ with $p/n\to y\in(0,1)$), showing that, after a specific normalization, the spectrum obeys the classical Marčenko–Pastur law, which is the well-known limit for sample covariance matrices. Extensions of this result are proven for general independent component models using analytic techniques involving the Stieltjes transform, and the theoretical claims are substantiated by extensive simulation studies (Wu, 24 Oct 2025).

1. Definition and Construction of $K_n$

The multivariate Kendall-$\tau$ matrix is constructed as follows: $$K_n = \frac{2}{n(n-1)} \sum_{i<j} \frac{ (X_i - X_j)(X_i - X_j)^\top }{ \|X_i - X_j\|^2 }$$ where $X_i \in \mathbb{R}^p$ are data vectors, and the normalization multiplies $K_n$ by $1/2$ per coordinate: $\tilde{K}_n = \frac{1}{2}p K_n$ The ESD of $\tilde{K}_n$ is the probability measure

$$F^{\tilde{K}_n}(x) = \frac{1}{p} \#\{ \lambda_i : \lambda_i \le x \}$$

where $\lambda_1,\ldots,\lambda_p$ are the eigenvalues of $\tilde{K}_n$.

2. Limiting Spectral Distribution: Marčenko–Pastur Law

Under the regime $p/n \to y \in (0,1)$, the ESD of $\tilde{K}_n$ converges almost surely to the Marčenko–Pastur law with variance parameter $1/2$: $$f_y(x) = \frac{1}{2\pi x y} \sqrt{ (b - x)(x - a) } \quad \text{for} \ x\in[a, b]; \quad 0 \text{ otherwise}$$ with

$$a = \frac{1}{2}(1 - \sqrt{y})^2 \,,\quad b = \frac{1}{2}(1 + \sqrt{y})^2$$

This result demonstrates that the limiting spectrum is identical (after normalization) to the corresponding limit for conventional sample covariance matrices, despite Kendall-$\tau$ being a nonlinear, rank-based operator.

Table: Marčenko–Pastur Law for $\frac{1}{2}pK_n$

Parameter	Expression	Range
Support	$[a, b]$	$a < x < b$
Density	$f_y(x)$	See above
Edge Points	$$a = \frac{1}{2}(1 - \sqrt{y})^2,\ b = \frac{1}{2}(1 + \sqrt{y})^2$$	$y \in (0,1)$

3. Stieltjes Transform Analysis

The proof leverages the Stieltjes transform, which for Hermitian $A$ is defined as

$$m_A(z) = \frac{1}{p} \operatorname{tr}\left( (A - zI)^{-1} \right), \qquad z \in \mathbb{C}^+$$

The limiting behavior of the Stieltjes transform of $\tilde{K}_n$ matches that of the Marčenko–Pastur distribution. The approach establishes that the normalized resolvent of the Kendall-$\tau$ matrix, when $p, n \to \infty$ with $p/n \to y$, satisfies the fixed-point equation

$m(z) = \frac{1}{ -z + \frac{y}{1 + m(z)} }$

appropriate to the variance-normalized Marčenko–Pastur law.

The methodology is robust to extensions involving more general data-generation models and facilitates analytic characterization of the LSD under additional structural conditions.

4. Generalization: Independent Component Model and Fixed-Point Equation

The results are extended to the independent component model with

$X_i = \mu + \Sigma^{1/2} Z_i$

where $Z_i$ have independent entries with mean zero, variance $1/2$, and finite fourth moment, and $\Sigma$ is symmetric positive definite with empirical spectral measure converging to $H$. In this setting, for the normalized statistic $\frac{1}{2}\text{tr}\,\Sigma K_n$, the limiting Stieltjes transform $m_F(z)$ is characterized by the fixed-point integral equation: $$m_F(z) = \int \frac{1}{ \tau(1-y - y z m_F(z)) - z }\, dH(\tau)$$ When $\Sigma$ is identity, $H$ is a Dirac mass at $1/2$ and the classical result for the Marčenko–Pastur law is recovered.

5. Simulation Validation

Monte Carlo simulations quantitatively corroborate the convergence of the ESD of $\frac{1}{2}pK_n$ to the Marčenko–Pastur density:

• Simulated data from standard normal, scaled normal, and uniform distributions with various aspect ratios ($p/n=0.25$, $0.5$, $0.75$)
• For moderate sample sizes, integrated squared errors (ISE) between empirical and theoretical densities are $<10^{-3}$
• Simulations further confirm the spectral invariance of $K_n$ under scale transformations. That is, scaling all $X_i$ leaves the spectrum unchanged.

6. Significance, Robustness, and Applications

A noteworthy attribute is that the Kendall-$\tau$ spectrum shares its universality class with sample covariance matrices, providing robust spectral analysis under non-Gaussian or heavy-tailed data. The convergence to the Marčenko–Pastur law validates the use of eigenvalue-based techniques (such as principal component analysis and subspace estimation) in high-dimensional settings for robust Kendall-type matrices, and confirms asymptotic stability with respect to scale transformations.

The extension to independent component models establishes the analytic tractability of spectral methods for broader classes of elliptical and non-elliptical ensembles, with explicit limiting equations for the Stieltjes transform.

7. Connections to Related Random Matrix Ensembles

Despite its construction from volatile nonlinear pairwise statistics, $K_n$ exhibits spectral limits paralleling linear operators (as seen in sample covariance matrices). The use of fixed-point equations for the Stieltjes transform is analogous to classical random matrix ensembles, further enabling theoretical analyses and methodological developments for robust, rank-based high-dimensional inference. The Marčenko–Pastur law continues to serve as the universal reference in this context, providing the spectral template against which robust estimators are benchmarked.