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Limiting Spectral Distribution of Large Sample Covariance Matrices Associated with a Class of Stationary Processes

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  • Marwa Banna

    (Université Paris Est)

  • Florence Merlevède

    (Université Paris Est)

Abstract

In this paper, we derive an extension of the Marc̆enko–Pastur theorem to a large class of weak dependent sequences of real-valued random variables having only moment of order 2. Under a mild dependence condition that is easily verifiable in many situations, we derive that the limiting spectral distribution of the associated sample covariance matrix is characterized by an explicit equation for its Stieltjes transform, depending on the spectral density of the underlying process. Applications to linear processes, functions of linear processes, and ARCH models are given.

Suggested Citation

  • Marwa Banna & Florence Merlevède, 2015. "Limiting Spectral Distribution of Large Sample Covariance Matrices Associated with a Class of Stationary Processes," Journal of Theoretical Probability, Springer, vol. 28(2), pages 745-783, June.
    • Handle: RePEc:spr:jotpro:v:28:y:2015:i:2:d:10.1007_s10959-013-0508-x
    DOI: 10.1007/s10959-013-0508-x
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    References listed on IDEAS

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    1. Giraitis, Liudas & Kokoszka, Piotr & Leipus, Remigijus, 2000. "Stationary Arch Models: Dependence Structure And Central Limit Theorem," Econometric Theory, Cambridge University Press, vol. 16(1), pages 3-22, February.
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    4. Cheng Wang & Baisuo Jin & Baiqi Miao, 2011. "On limiting spectral distribution of large sample covariance matrices by VARMA(p,q)," Journal of Time Series Analysis, Wiley Blackwell, vol. 32(5), pages 539-546, September.
    5. Silverstein, J. W. & Bai, Z. D., 1995. "On the Empirical Distribution of Eigenvalues of a Class of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 54(2), pages 175-192, August.
    6. Pan, Guangming, 2010. "Strong convergence of the empirical distribution of eigenvalues of sample covariance matrices with a perturbation matrix," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1330-1338, July.
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    Cited by:

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    3. Sanders, Jaron & Van Werde, Alexander, 2023. "Singular value distribution of dense random matrices with block Markovian dependence," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 453-504.

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