Covariance Matrix Estimation from Linearly-Correlated Gaussian Samples

Covariance matrix estimation concerns the problem of estimating the covariance matrix from a collection of samples, which is of extreme importance in many applications. Classical results have shown that O(n) samples are sufficient to accurately estimate the covariance matrix from n-dimensional independent Gaussian samples. However, in many practical applications, the received signal samples might be correlated, which makes the classical analysis inapplicable. In this paper, we develop a nonasymptotic analysis for the covariance matrix estimation from linearly-correlated Gaussian samples. Our theoretical results show that the error bounds are determined by the signal dimension n, the sample size m, and the shape parameter of the distribution of the correlated sample covariance matrix. Particularly, when the shape parameter is a class of Toeplitz matrices (which is of great practical interest), O(n) samples are also sufficient to faithfully estimate the covariance matrix from correlated samples. Simulations are provided to verify the correctness of the theoretical results.