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

TL;DR

This paper proves a central limit theorem for linear spectral statistics of large separable sample covariance matrices, showing they tend to a Gaussian distribution as the matrix dimensions grow proportionally.

Contribution

It establishes the Gaussian limit for LSS of separable sample covariance matrices under general conditions, extending classical results to a broader matrix class.

Findings

LSS of large separable covariance matrices are asymptotically Gaussian.

The result holds when the ratio of dimensions converges to a positive constant.

The theorem applies to matrices with independent entries having mean zero, variance one, and fourth moment three.

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