The asymptotics of eigenvector overlaps of large sample covariance matrices and nonlinear shrinkage estimator

Consider the data matrix Y=(y_1,...,y_n). Denote the left and right singular vectors of Y by u_i and v_j respectively. This talk discusses the asymptotic behaviour of the eigenvector/singular vector overlaps u_i^TD_1u_j v_i^TD_2v_j and u_i^TD_3v_j. We establish the convergence in probability of these eigenvector overlaps toward their deterministic counterparts with explicit convergence rates when the dimension and the sample size are proportional to each other. Relying on this, we report a more precise characterization of the loss for Ledoit and Wolf's nonlinear shrinkage estimators of the population covariance matrix.