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

Abstract: Consider the sample covariance matrix $$\Sigma^{1/2}XX^T\Sigma^{1/2}$$ 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 $$\sum_if(\lambda_i)$$ converges in distribution to a Gaussian distribution. Here ${\lambda_i}$ are eigenvalues of $\Sigma^{1/2}XX^T\Sigma^{1/2}$ 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 $$\frac{1}{\sqrt N}\sum_if(\lambda_i)$$ converges in distribution to a Gaussian distribution. This phenomenon implies that the randomness of $\Sigma$ decreases the correlation among ${\lambda_i}$.