# The Local Ledoit-Peche Law

## TL;DR

This paper refines the Ledoit-Peche law by establishing an optimal convergence rate for functions of random covariance matrices, with implications for improved shrinkage covariance estimation.

## Contribution

It provides an essentially optimal convergence rate for the Ledoit-Peche law, advancing understanding of covariance matrix estimators in high-dimensional statistics.

## Key findings

- Established an optimal convergence rate for the Ledoit-Peche law
- Hypothesized the rate to be the minimal possible for MV loss
- Implications for improved shrinkage covariance estimation

## Abstract

Ledoit and Peche proved convergence of certain functions of a random covariance matrix's resolvent; we refer to this as the Ledoit-Peche law. One important application of their result is shrinkage covariance estimation with respect to so-called Minimum Variance (MV) loss, discussed in the work of Ledoit and Wolf. We provide an essentially optimal rate of convergence and hypothesize it to be the smallest possible rate of excess MV loss within the shrinkage class.