QUANTITATIVE CLT FOR LINEAR EIGENVALUE STATISTICS OF WIGNER MATRICES

In this article, we establish a near-optimal convergence rate for the CLT of linear eigenvalue statistics of N x N Wigner matrices, in Kolmogorov- Smirnov distance. For all test functions f is an element of C5(R), we show that the conver-gence rate is either N-1/2+epsilon or N-1+epsilon, depending on the first Chebyshev coefficient of f and the third moment of the diagonal matrix entries. The condition that distinguishes these two rates is necessary and sufficient. For a general class of test functions, we further identify matching lower bounds for the convergence rates. In addition, we identify an explicit, nonuniversal con-tribution in the linear eigenvalue statistics, which is responsible for the slow rate N-1/2+epsilon for non-Gaussian ensembles. By removing this nonuniversal part, we show that the shifted linear eigenvalue statistics have the unified convergence rate N-1+epsilon for all test functions.