Covariance estimation with direction dependence accuracy

We construct an estimator $\widehatΣ$ for covariance matrices of unknown, centred random vectors X, with the given data consisting of N independent measurements $X_1,...,X_N$ of X and the wanted confidence level. We show under minimal assumptions on X, the estimator performs with the optimal accuracy with respect to the operator norm. In addition, the estimator is also optimal with respect to direction dependence accuracy: $\langle \widehatΣu,u\rangle$ is an optimal estimator for $σ^2(u)=\mathbb{E}\langle X,u\rangle^2$ when $σ^2(u)$ is ``large".