Fluctuations for linear eigenvalue statistics of sample covariance random matrices

Giorgio Cipolloni, László Erdős

Published 2018-06-22Version 1

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 the fluctuations of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of $\widetilde{W}$ and $W$. Unlike in a similar result for Wigner matrices [arXiv:1608.05163], for sample covariance matrices the fluctuation may entirely vanish and we identify these cases.