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.
Categories: math.PR