The Local Ledoit-Peche Law

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.