# Eigenvectors of Some Large Sample Covariance Matrix
Ensembles

## Olivier Ledoit and Sandrine Péché

### Abstract

We consider sample covariance matrices S_{N} = (1/p)
Σ_{N}^{1/2}X_{N}X_{N}^{*}Σ_{N}^{1/2}
where X_{N} is a N × p
real or complex matrix with i.i.d. entries with finite 12^{th} 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 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(Σ_{N})(S_{N}-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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