 # Eigenvectors of some large sample covariance matrix ensembles

Ledoit, Olivier; Péché, Sandrine (2011). Eigenvectors of some large sample covariance matrix ensembles. Probability Theory and Related Fields, 151(1-2):233-264.

## Abstract

We consider sample covariance matrices $${S_N=\frac{1}{p}\Sigma_N^{1/2}X_NX_N^* \Sigma_N^{1/2}}$$ where X N 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 → ∞ and p/N → γ >0. We quantify the relationship between sample and population eigenvectors by studying the asymptotics of functionals of the type $${\frac{1}{N}\text{Tr} ( g(\Sigma_N) (S_N-zI)^{-1}),}$$ 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

## Abstract

We consider sample covariance matrices $${S_N=\frac{1}{p}\Sigma_N^{1/2}X_NX_N^* \Sigma_N^{1/2}}$$ where X N 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 → ∞ and p/N → γ >0. We quantify the relationship between sample and population eigenvectors by studying the asymptotics of functionals of the type $${\frac{1}{N}\text{Tr} ( g(\Sigma_N) (S_N-zI)^{-1}),}$$ 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

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Item Type: Journal Article, refereed, original work National licences > 142-005 510 Mathematics Physical Sciences > Analysis Physical Sciences > Statistics and Probability Social Sciences & Humanities > Statistics, Probability and Uncertainty English 1 October 2011 05 Dec 2018 19:10 31 Jul 2020 02:37 Springer 0178-8051 Green https://doi.org/10.1007/s00440-010-0298-3 https://www.swissbib.ch/Search/Results?lookfor=nationallicencespringer101007s0044001002983 (Library Catalogue)

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