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Eigenvectors of some large sample covariance matrix ensembles
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  • Published: 06 May 2010

Eigenvectors of some large sample covariance matrix ensembles

  • Olivier Ledoit1 &
  • Sandrine Péché2 

Probability Theory and Related Fields volume 151, pages 233–264 (2011)Cite this article

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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 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 \({\frac{1}{N}\text{Tr} ( g(\Sigma_N) (S_N-zI)^{-1}),}\) 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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Author information

Authors and Affiliations

  1. Institute for Empirical Research in Economics, University of Zurich, Blümlisalpstrasse 10, 8006, Zurich, Switzerland

    Olivier Ledoit

  2. Institut Fourier, Université Grenoble 1, 100 rue des Maths, BP 74, 38402, Saint-Martin-d’Hères, France

    Sandrine Péché

Authors
  1. Olivier Ledoit
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  2. Sandrine Péché
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Corresponding author

Correspondence to Olivier Ledoit.

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Cite this article

Ledoit, O., Péché, S. Eigenvectors of some large sample covariance matrix ensembles. Probab. Theory Relat. Fields 151, 233–264 (2011). https://doi.org/10.1007/s00440-010-0298-3

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  • Received: 24 March 2009

  • Revised: 10 March 2010

  • Published: 06 May 2010

  • Issue Date: October 2011

  • DOI: https://doi.org/10.1007/s00440-010-0298-3

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Keywords

  • Asymptotic distribution
  • Bias correction
  • Eigenvectors and eigenvalues
  • Principal component analysis
  • Random matrix theory
  • Sample covariance matrix
  • Shrinkage estimator
  • Stieltjes transform

Mathematics Subject Classification (2000)

  • 60B20
  • 62J10
  • 15B52
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