Eigenvectors of some large sample covariance matrix ensembles

• Published in: Probability theory and related fields Vol. 151; no. 1-2; pp. 233 - 264
• Main Authors: Ledoit, Olivier, Péché, Sandrine
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.10.2011; Springer; Springer Nature B.V
• Subjects: Asymptotic methods; Asymptotic properties; Bias; Complex numbers; Covariance matrix; Distribution theory; Economics; Eigenvalues; Eigenvectors; Exact sciences and technology; Finance; General topics; Insurance; Limit theorems; Management; Mathematical analysis; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Matrix; Neighborhoods; Normal distribution; Operations Research/Decision Theory; Population; Principal components analysis; Probability; Probability and statistics; Probability distribution; Probability Theory and Stochastic Processes; Quantitative Finance; Random variables; Sciences and techniques of general use; Statistics; Statistics for Business; Studies; Theoretical; Sample covariance matrix; 15B52; Bias correction; Random matrix theory; Eigenvectors and eigenvalues; Shrinkage estimator; 60B20; Asymptotic distribution; 62J10; Principal component analysis; Stieltjes transform; Random matrix; Bias; Large sample; Asymptotic optimality; Eigenvalue; Probability distribution; Limit distribution; Inverse matrix; Real matrix; Statistical moment; Eigenvector; Probability theory; Covariance matrix; Matrix function; Functional; Positive definite matrix; Biased estimation
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-010-0298-3

We consider sample covariance matrices where X N is a N ×  p real or complex matrix with i.i.d. entries with finite 12th moment and Σ N is a N ×  N positive definite matrix. In addition we assume that the spectral measure of Σ N almost surely converges to some limiting probability distribution as N → ∞ and p / N → γ > 0. We quantify the relationship between sample and population eigenvectors by studying the asymptotics of functionals of the type where I is the identity matrix, g is a bounded function and z is a complex number. This is then used to compute the asymptotically optimal bias correction for sample eigenvalues, paving the way for a new generation of improved estimators of the covariance matrix and its inverse.