  In this work we construct an optimal linear shrinkage estimator for the
covariance matrix in high dimensions. The recent results from the random matrix
theory allow us to find the asymptotic deterministic equivalents of the optimal
shrinkage intensities and estimate them consistently. The developed
distribution-free estimators obey almost surely the smallest Frobenius loss
over all linear shrinkage estimators for the covariance matrix. The case we
consider includes the number of variables $p\rightarrow\infty$ and the sample
size $n\rightarrow\infty$ so that $p/n\rightarrow c\in (0, +\infty)$.
Additionally, we prove that the Frobenius norm of the sample covariance matrix
tends almost surely to a deterministic quantity which can be consistently
estimated.
