A short proof of Ledoit-P\'ech\'e's RIE formula for covariance matrices

This is a short proof of Ledoit-P'ech'e’s RIE formula for covariance matrices. The proof is based on the Stein formula, which gives a very simple way to derive the result. One of the advantages of this approach is that it shows that the only really needed hypothesis, for the machinery to work, is that the mean of the eigenvalues of the true covariance matrix and the largest of them have the same order.