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

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 $\mathbf{B}_n = \frac1N\mathbf{T}_{2n}^{1/2} \mathbf{X}_n \mathbf{T}_{1n} \mathbf{X}_n' \mathbf{T}_{2n}^{1/2}$ 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.