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

Abstract

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