# Convergence of eigenvector empirical spectral distribution of sample covariance matrices

## TL;DR

This paper establishes improved convergence rates for the eigenvector empirical spectral distribution of sample covariance matrices to the deformed Marčenko-Pastur law, under weaker moment conditions and more general matrix models.

## Contribution

It provides sharper convergence rate bounds for VESD to the deformed MP distribution, extending previous results to broader settings with weaker assumptions.

## Key findings

- Expected VESD converges to deformed MP law at rate N^{-1+ε}.
- Almost sure convergence rate of VESD is improved to N^{-1/2+ε}.
- Results hold under finite 6th and 8th moment conditions, with general covariance matrices.

## Abstract

The eigenvector empirical spectral distribution (VESD) is a useful tool in studying the limiting behavior of eigenvalues and eigenvectors of covariance matrices. In this paper, we study the convergence rate of the VESD of sample covariance matrices to the deformed Mar\v{c}enko-Pastur (MP) distribution. Consider sample covariance matrices of the form $\Sigma^{1/2} X X^* \Sigma^{1/2}$, where $X=(x_{ij})$ is an $M\times N$ random matrix whose entries are independent random variables with mean zero and variance $N^{-1}$, and $\Sigma$ is a deterministic positive-definite matrix. We prove that the Kolmogorov distance between the expected VESD and the deformed MP distribution is bounded by $N^{-1+\epsilon}$ for any fixed $\epsilon>0$, provided that the entries $\sqrt{N}x_{ij}$ have uniformly bounded 6th moments and $|N/M-1|\ge \tau$ for some constant $\tau>0$. This result improves the previous one obtained in \cite{XYZ2013}, which gave the convergence rate $O(N^{-1/2})$ assuming $i.i.d.$ $X$ entries, bounded 10th moment, $\Sigma=I$ and $M<N$. Moreover, we also prove that under the finite $8$th moment assumption, the convergence rate of the VESD is $O(N^{-1/2+\epsilon})$ almost surely for any fixed $\epsilon>0$, which improves the previous bound $N^{-1/4+\epsilon}$ in \cite{XYZ2013}.