Covariance Estimation under Missing Observations and $L_4-L_2$ Moment Equivalence

We consider the problem of estimating the covariance matrix of a random vector by observing i.i.d samples and each entry of the sampled vector is missed with probability $p$. Under the standard $L_4-L_2$ moment equivalence assumption, we construct the first estimator that simultaneously achieves optimality with respect to the parameter $p$ and it recovers the optimal convergence rate for the classical covariance estimation problem when $p=1$