Quantitative CLT for linear eigenvalue statistics of Wigner matrices
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In this article, we establish a near-optimal convergence rate for the CLT of linear eigenvalue statistics of Wigner matrices, in Kolmogorov-Smirnov distance. For all test functions f∈ C^5(ℝ), we show that the convergence rate is either N^-1/2+ε or N^-1+ε, 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, non-universal contribution in the linear eigenvalue statistics, which is responsible for the slow rate N^-1/2+ε for non-Gaussian ensembles. By removing this non-universal part, we show that the shifted linear eigenvalue statistics have the unified convergence rate N^-1+ε for all test functions.
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