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 $N^{-1/2+\varepsilon}$ or $N^{-1+\varepsilon}$, 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 $N^{-1/2+\varepsilon}$ for non-Gaussian ensembles. By removing this non-universal part, we show that the shifted linear eigenvalue statistics have the unified convergence rate $N^{-1+\varepsilon}$ for all test functions.
翻译：在本篇文章中,我们用科尔莫戈洛夫-斯米尔诺夫距离为维格勒矩阵的线性电子值统计CLT设定了接近最佳的趋同率。对于所有测试函数, $f\ in C5 (\ mathbb R) $, 我们显示, 趋同率要么是 $N ⁇ -1/2 ⁇ varepsilon} $, 要么是 $ ⁇ -1 ⁇ -1 ⁇ ⁇ ⁇ varepsilon} $, 取决于第一个Chebyshev 系数($ f) 和对角矩阵条目的第三个时刻。 区分这两个比率的条件既必要又充分。 对于一般的测试函数, 我们进一步确定匹配趋同率的下限。 此外, 我们在线性电子值统计中确定一个明确的非通用贡献值, 导致非加萨西南方币的低速率 $ ⁇ -1/2 ⁇ varepsilon}。 通过删除这一非普遍性部分, 我们显示, 转移的线性电子值统计具有所有测试功能的统一趋同率 $N ⁇ -1 ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ } 。
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