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

Abstract

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 ${\mathbf{B}}_n=\frac{1}{n}{\mathbf{T}}_p^{1/2}{\mathbf{X}}_n{\mathbf{X}}_n^{\ast }{\mathbf{T}}_p^{1/2},$ where ${\mathbf{X}}_n=(x_{ij})$ 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 $\kappa >0$.

Modified some typos and added some references