标题： 关于大维样本协方差矩阵LSS的中心极限定理收敛速度的研究 显示英文标题

标题： On the rate of convergence in the CLT for LSS of large-dimensional sample covariance matrices

摘要： 本文研究了与大维样本协方差矩阵相关的线性谱统计量（LSS）的中心极限定理的收敛速度。我们考虑形式为${\mathbf B}_n=\frac{1}{n}{\mathbf T}_p^{1/2}{\mathbf X}_n{\mathbf X}_n^*{\mathbf T}_p^{1/2},$的矩阵，其中${\mathbf X}_n= (x_{i j} ) $是一个$p \times n$阶的矩阵，其元素是独立同分布（i.i.d.）的实值或复数值变量，而${\mathbf T} _p$是一个$p\times p$阶的非随机Hermitian半正定矩阵，并且其谱范数在$p$中一致有界。 利用Stein方法，我们证明了如果矩阵的元素 $x_{ij}$ 满足条件 $\mathbb{E}|x_{ij}|^{10}<\infty$，并且维数与样本量之比 $p/n\to y>0$ 趋于 $n\to\infty$，则对于固定的 $\kappa>0$， ${\mathbf B}_n$ 的标准化LSS到标准正态分布的Kolmogorov-Smirnov距离的收敛速度为 $O(n^{-1/2+\kappa})$。 显示英文摘要

摘要： This paper investigates the rate of convergence for the central limit theorem of linear spectral statistic (LSS) associated with large-dimensional sample covariance matrices. We consider matrices of the form ${\mathbf B}_n=\frac{1}{n}{\mathbf T}_p^{1/2}{\mathbf X}_n{\mathbf X}_n^*{\mathbf T}_p^{1/2},$ where ${\mathbf X}_n= (x_{i j} ) $ is a $p \times n$ matrix whose entries are independent and identically distributed (i.i.d.) real or complex variables, and ${\mathbf T} _p$ is a $p\times p$ nonrandom Hermitian nonnegative definite matrix with its spectral norm uniformly bounded in $p$. Employing Stein's method, we establish that if the entries $x_{ij}$ satisfy $\mathbb{E}|x_{ij}|^{10}<\infty$ and the ratio of the dimension to sample size $p/n\to y>0$ as $n\to\infty$, then the convergence rate of the normalized LSS of ${\mathbf B}_n$ to the standard normal distribution, measured in the Kolmogorov-Smirnov distance, is $O(n^{-1/2+\kappa})$ for any fixed $\kappa>0$.