Eigenvector overlaps in large sample covariance matrices and nonlinear shrinkage estimators

Consider a data matrix Y = y₁, , yN of size M N, where the columns are independent observations from a random vector y with zero mean and population covariance. Let uᵢ and vⱼ denote the left and right singular vectors of Y, respectively. This study investigates the eigenvector/singular vector overlaps uᵢ, D₁ uⱼ, vᵢ, D₂ vⱼ and uᵢ, D₃ vⱼ, where Dₖ are general deterministic matrices with bounded operator norms. We establish the convergence in probability of these eigenvector overlaps toward their deterministic counterparts with explicit convergence rates, when the dimension M scales proportionally with the sample size N. Building on these findings, we offer a more precise characterization of the loss for Ledoit and Wolf's nonlinear shrinkage estimators of the population covariance.