How Close Is the Sample Covariance Matrix to the Actual Covariance Matrix?

Given a probability distribution inRn with general (non-white) covariance, a classical estimator of the covariance matrix is the sample covariance matrix obtained from a sample of N independent points. What is the optimal sample size N = N(n) that guarantees estimation with a fixed accuracy in the operator norm? Suppose the distribution is supported in a centered Euclidean ball of radius O( √ n). We conjecture that the optimal sample size is N = O(n) for all distributions with finite fourth moment, and we prove this up to an iterated logarithmic factor. This problem is motivated by the optimal theorem of M. Rudelson [23] which states that N = O(n logn) for distributions with finite second moment, and a recent result of R. Adamczak et al. [1] which guarantees that N = O(n) for sub-exponential distributions.