Central limit theorem for linear spectral statistics of large dimensional quaternion sample covariance matrices (Q725526)

The subject of this paper is a central limit theorem for the linear spectral statistics \(1/n\, \Sigma_{j=1}^nf(\lambda_j^n)\), where \(f\) is a test function and \(\lambda_1^n, \ldots, \lambda_n^n\) are the eigenvalues of a matrix \({A}_n\). Here, \({A}_n\) is a quaternion sample covariance matrix of type \(1/n\, {\Sigma}_n^{1/2}{X}_n^{\ast}{X}_n {\Sigma}_n^{1/2}\) where \({X}_n\) is a \(p\times n\) quaternion random matrix and \({\Sigma}_n\) a \(p\times p\) quaternion Hermitian matrix. The Wishart quaternion matrices are particular cases of such covariance matrices.