Covariance Matrix Estimation From Linearly-Correlated Gaussian Samples

Abstract

Covariance matrix estimation concerns the problem of estimating the\ncovariance matrix from a collection of samples, which is of extreme importance\nin many applications. Classical results have shown that $O(n)$ samples are\nsufficient to accurately estimate the covariance matrix from $n$-dimensional\nindependent Gaussian samples. However, in many practical applications, the\nreceived signal samples might be correlated, which makes the classical analysis\ninapplicable. In this paper, we develop a non-asymptotic analysis for the\ncovariance matrix estimation from correlated Gaussian samples. Our theoretical\nresults show that the error bounds are determined by the signal dimension $n$,\nthe sample size $m$, and the shape parameter of the distribution of the\ncorrelated sample covariance matrix. Particularly, when the shape parameter is\na class of Toeplitz matrices (which is of great practical interest), $O(n)$\nsamples are also sufficient to faithfully estimate the covariance matrix from\ncorrelated samples. Simulations are provided to verify the correctness of the\ntheoretical results.\n