Quantitative CLT for linear eigenvalue statistics of Wigner matrices

Abstract

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 $f\in C^5(\mathbb{R})$, we show that the convergence 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, non-universal contribution in the linear eigenvalue statistics, which is responsible for the slow rate $N^{-1/2+\epsilon }$ for non-Gaussian ensembles. By removing this non-universal part, we show that the shifted linear eigenvalue statistics have the unified convergence rate $N^{-1+\epsilon }$ for all test functions.

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References in corpus (4)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices
• Second Order Freeness and Fluctuations of Random Matrices: II. Unitary Random Matrices
• Second Order Freeness and Fluctuations of Random Matrices, III. Higher order freeness and free cumulants

Cited by in corpus (1)

• Fluctuations of the Stieltjes transform of the empirical spectral distribution of selfadjoint polynomials in Wigner and deterministic diagonal matrices