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Eigenvalues Distribution Limit of Covariance Matrices with AR Processes Entries

  • Zahira KhettabEmail author
  • Tahar Mourid
Conference paper
  • 1.1k Downloads
Part of the Contributions to Statistics book series (CONTRIB.STAT.)

Abstract

We consider a class of random matrices \(B_{N}=X_{N}T_{N}X_{N}^{t},\) where \( X_{N}\) is a matrix \((N\times n(N))\) whose rows are independent, the entries \(X_{ij}\) in each row satisfy an autoregressive relation AR(1), and \( T_{N}\) is a diagonal matrix independent of \(X_{N}\). Under some conditions, we show that if the empirical distribution function of eigenvalues of \(T_{N}\) converges almost surely to a proper probability distribution as \(N\longrightarrow \infty \) and \(\frac{n(N)}{N}\longrightarrow c>0\), then the empirical distribution function of eigenvalues of \(B_{N}\) converges almost surely to a non-random limit function given by Marcenko and Pastur. Numerical simulations illustrate the behavior of kernel density estimators and density estimators of Stieltjes transform around the true density and we give a numerical comparison on the base of \(L_{1}\) error varying different parameters.

Keywords

Large dimensional random matrix Empirical distribution function of eigenvalues Covariance matrix Autoregressive processes Stieltjes transform Kernel density estimators 

Notes

Acknowledgements

The authors would like to thank the Editor and anonymous referees for insightful comments improving the presentation of this paper.

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Laboratoire de Statistiques et Modélisations AléatoiresUniversity of TlemcenTlemcenAlgeria

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