Asymptotics of empirical eigenvalues for large separable covariance matrices

We investigate the asymptotics of eigenvalues of sample covariance matrices associated with a class of non-independent Gaussian processes (separable and temporally stationary) under the Kolmogorov asymptotic regime. The limiting spectral distribution (LSD) is shown to depend explicitly on the Kolmogorov constant (a fixed limiting ratio of the sample size to the dimensionality) and parameters representing the spatio- and temporal- correlations. The Cauchy, M- and N-transforms from free harmonic analysis play key roles to this LSD calculation problem. The free multiplication law of free random variables is employed to give a semi-closed-form expression (only the final step is numerical based) of the LSD for the spatio-covariance matrix being a diagonally dominant Wigner matrix and temporal-covariance matrix an exponential off-diagonal decay (Toeplitz) matrix. Furthermore, we also derive a nonlinear shrinkage estimator for the top eigenvalues associated with a low rank matrix (Hermitian) from its noisy measurements. Numerical studies about the effectiveness of the estimator are also presented.