The paper proves several limit theorems for linear eigenvalue statistics of overlapping Wigner and sample covariance matrices. It is shown that the covariance of the limiting multivariate Gaussian distribution is diagonalized by choosing the Chebyshev polynomials of the first kind as the basis for the test function space. The covariance is explicitly computed and it is shown that for the Chebyshev polynomials of sufficiently high degree the covariance of linear statistics depends only on the first two moments of matrix entries.
"Limit theorems for linear eigenvalue statistics of overlapping matrices." Electron. J. Probab. 20 1 - 30, 2015. https://doi.org/10.1214/EJP.v20-3937