Central limit theorem for eigenvalue statistics of sample covariance matrix with random population

Abstract

Consider the sample covariance matrix $${\Sigma }^{1/2}X{X}^T{\Sigma }^{1/2}$$ where $X$ is an $M\times N$ random matrix with independent entries and $\Sigma$ is an $M\times M$ diagonal matrix. It is known that if $\Sigma$ is deterministic, then the fluctuation of $$\sum _{i}f({\lambda }_i)$$ converges in distribution to a Gaussian distribution. Here $\{{\lambda }_i\}$ are eigenvalues of ${\Sigma }^{1/2}X{X}^T{\Sigma }^{1/2}$ and $f$ is a good enough test function. In this paper we consider the case that $\Sigma$ is random and show that the fluctuation of $$\frac{1}{\sqrt{N}}\sum _{i}f({\lambda }_i)$$ converges in distribution to a Gaussian distribution. This phenomenon implies that the randomness of $\Sigma$ decreases the correlation among $\{{\lambda }_i\}$.

Errors corrected. More details included. Main theorem strengthened