Fluctuations for linear eigenvalue statistics of sample covariance matrices

TL;DR

This paper establishes a central limit theorem for the difference in linear eigenvalue statistics of a sample covariance matrix and its minor, revealing smaller fluctuations due to strong eigenvalue correlations, with implications for Gaussian field derivatives.

Contribution

It provides the first CLT for the eigenvalue statistic difference of sample covariance matrices and their minors, highlighting unique fluctuation behaviors compared to Wigner matrices.

Findings

Fluctuations of the difference are significantly smaller than individual statistics.

Eigenvalue correlations lead to potential vanishing fluctuations.

Results connect to the Gaussian field derivatives in recent literature.

Abstract

We prove a central limit theorem for the difference of linear eigenvalue statistics of a sample covariance matrix $\widetilde{W}$ and its minor $W$. We find that the fluctuation of this difference is much smaller than those of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of $\widetilde{W}$ and $W$. Our result identifies the fluctuation of the spatial derivative of the approximate Gaussian field in the recent paper by Dumitru and Paquette. Unlike in a similar result for Wigner matrices, for sample covariance matrices the fluctuation may entirely vanish.