Eigenvectors of Some Large Sample Covariance Matrix Ensembles

Olivier Ledoit and Sandrine Péché


We consider sample covariance matrices SN = (1/p) ΣN1/2XNXN*ΣN1/2 where XN is a N × p real or complex matrix with i.i.d. entries with finite 12th moment and ΣN is a N × N positive definite matrix. In addition we assume that the spectral measure of ΣN almost surely converges to some limiting probability distribution as N goes to infinity and p/N converges to γ > 0.We quantify the relationship between sample and population eigenvectors by studying the asymptotics of functionals of the type (1/N)Tr(g(ΣN)(SN-zI)-1), 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.

Probability Theory and Related Fields, Volume 151, nos. 1-2 (2011), pages 233-264.

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