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 ${\mathbf{T}}_{2n}^{1\mathrm{/}2}$ 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 $n\mathrm{/}N$ approaches a positive constant.