For generalized Wigner matrices, the characteristic function of linear spectral statistics is expanded around its Gaussian limit with error O(N^{-1}) and an explicit N^{-1/2} correction.
Central Limit Theorem for Linear Statistics of Eigenvalues of Band Random Matrices
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We prove the Central Limit Theorem for linear statistics of the eigenvalues of band random matrices provided $\sqrt{n} \ll b_n \ll n$ and test functions are sufficiently smooth.
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Some estimates for generalized Wigner matrix linear spectral statistics
For generalized Wigner matrices, the characteristic function of linear spectral statistics is expanded around its Gaussian limit with error O(N^{-1}) and an explicit N^{-1/2} correction.