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

TL;DR

This paper provides a concise proof of Ledoit-Péché's RIE formula for covariance matrices using Stein's formula, highlighting minimal assumptions needed for the result to hold.

Contribution

It introduces a simplified proof method for the RIE formula, relying only on basic eigenvalue order conditions.

Findings

The proof simplifies understanding of the RIE formula.

It demonstrates the minimal assumptions required for the formula's validity.

The approach is based on Stein's formula, making the proof more accessible.

Abstract

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.