Fluctuations for differences of linear eigenvalue statistics for sample covariance matrices

We prove a central limit theorem for the difference of linear eigenvalue statistics of a sample covariance matrix 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 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.