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

Given a probability distribution in ℝ n with general (nonwhite) 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 that the distribution is supported in a centered Euclidean ball of radius $O(\sqrt{n})$ . 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 Rudelson (J. Funct. Anal. 164:60–72, 1999), which states that N=O(nlog n) for distributions with finite second moment, and a recent result of Adamczak et al. (J. Am. Math. Soc. 234:535–561, 2010), which guarantees that N=O(n) for subexponential distributions.