Central limit theorem for linear spectral statistics of large dimensional separable sample covariance matrices

Abstract

Suppose that ${\mathbf{X}}_n=(x_{jk})$ is $N\times n$ whose elements are independent real variables with mean zero, variance 1 and the fourth moment equal to three. The separable sample covariance matrix is defined as ${\mathbf{B}}_n=\frac{1}{N}{\mathbf{T}}_{2n}^{1/2}{\mathbf{X}}_n{\mathbf{T}}_{1n}{\mathbf{X}}_n^{\prime }{\mathbf{T}}_{2n}^{1/2}$ where ${\mathbf{T}}_{1n}$ is a symmetric matrix and ${\mathbf{T}}_{2n}^{1/2}$ is a symmetric square root of the nonnegative definite symmetric matrix ${\mathbf{T}}_{2n}$. Its linear spectral statistics (LSS) are shown to have Gaussian limits when $n/N$ approaches a positive constant.

References in corpus (1)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries