Fluctuations of eigenvalues of patterned random matrices

In this article, we study the fluctuations of linear statistics of eigenvalues of circulant, symmetric circulant, reverse circulant, and Hankel matrices. We show that the linear spectral statistics of these matrices converge to the Gaussian distribution in total variation norm when the matrices are constructed using independent copies of a standard normal random variable. We also calculate the limiting variance of the linear spectral statistics for circulant, symmetric circulant, and reverse circulant matrices.In this article, we study the fluctuations of linear statistics of eigenvalues of circulant, symmetric circulant, reverse circulant, and Hankel matrices. We show that the linear spectral statistics of these matrices converge to the Gaussian distribution in total variation norm when the matrices are constructed using independent copies of a standard normal random variable. We also calculate the limiting variance of the linear spectral statistics for circulant, symmetric circulant, and reverse circulant matrices.