On asymptotics of eigenvectors of large sample covariance matrix

TL;DR

This paper studies the asymptotic behavior of eigenvectors of large sample covariance matrices, establishing their spectral distribution convergence and Gaussian limits under certain conditions, with implications for their distributional properties.

Contribution

It introduces a new spectral distribution approach for eigenvectors and proves Gaussian limits for spectral statistics, advancing understanding of eigenvector asymptotics in large matrices.

Findings

Empirical spectral distribution of eigenvectors converges to the same limit as equal-weight distribution.

Linear spectral statistics based on eigenvectors are asymptotically Gaussian.

Eigenvector matrices are nearly Haar distributed when the population covariance is a multiple of identity.

Abstract

Let \{$X_{ij}$\}, $i,j=...,$ be a double array of i.i.d. complex random variables with $EX_{11}=0,E|X_{11}|^2=1$ and $E|X_{11}|^4<\infty$, and let $A_n=\frac{1}{N}T_n^{{1}/{2}}X_nX_n^*T_n^{{1}/{2}}$, where $T_n^{{1}/{2}}$ is the square root of a nonnegative definite matrix $T_n$ and $X_n$ is the $n\times N$ matrix of the upper-left corner of the double array. The matrix $A_n$ can be considered as a sample covariance matrix of an i.i.d. sample from a population with mean zero and covariance matrix $T_n$, or as a multivariate $F$ matrix if $T_n$ is the inverse of another sample covariance matrix. To investigate the limiting behavior of the eigenvectors of $A_n$, a new form of empirical spectral distribution is defined with weights defined by eigenvectors and it is then shown that this has the same limiting spectral distribution as the empirical spectral distribution defined by equal weights. Moreover, if \{$X_{ij}$\} and $T_n$ are either real or complex and some additional moment assumptions are made then linear spectral statistics defined by the eigenvectors of $A_n$ are proved to have Gaussian limits, which suggests that the eigenvector matrix of $A_n$ is nearly Haar distributed when $T_n$ is a multiple of the identity matrix, an easy consequence for a Wishart matrix.