We give an upper bound on the total variation distance between the linear eigenvalue statistic, properly scaled and centered, of a random matrix with a variance profile and the standard Gaussian random variable. The second-order Poincaré inequality-type result introduced in [S. Chatterjee, Fluctuations of eigenvalues and second order poincaré inequalities, Prob. Theory Rel. Fields 143(1) (2009) 1–40.] is used to establish the bound. Using this bound, we prove central limit theorem for linear eigenvalue statistics of random matrices with different kind of variance profiles. We re-establish some existing results on fluctuations of linear eigenvalue statistics of some well-known random matrix ensembles by choosing appropriate variance profiles.

中文翻译：

---

具有方差分布的随机矩阵的线性特征值统计

我们给出了具有方差分布的随机矩阵和标准高斯随机变量的线性特征值统计量（适当缩放和居中）之间总变异距离的上限。[S. 中介绍的二阶庞加莱不等式结果。Chatterjee，特征值的波动和二阶庞加莱不等式，概率。理论相对。字段 143(1) (2009) 1-40.] 用于确定界限。使用这个界限，我们证明了具有不同类型方差分布的随机矩阵的线性特征值统计的中心极限定理。我们通过选择适当的方差分布重新建立了一些关于一些众所周知的随机矩阵集合的线性特征值统计波动的现有结果。