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

TL;DR

This paper extends the central limit theorem for eigenvalue statistics of sample covariance matrices to cases where the population covariance matrix is random, showing Gaussian fluctuations and reduced eigenvalue correlation.

Contribution

It demonstrates that when the population covariance matrix is random, the eigenvalue sum fluctuations still follow a Gaussian distribution, revealing the impact of covariance randomness.

Findings

Eigenvalue sum fluctuations are Gaussian with random covariance.

Randomness in covariance reduces eigenvalue correlation.

The fluctuation scale is 1/√N.

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 $\{\lambda_i\}$.