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On the empirical spectral distribution for matrices with long memory and independent rows

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  • Merlevède, F.
  • Peligrad, M.

Abstract

In this paper we show that the empirical eigenvalue distribution of any sample covariance matrix generated by independent samples of a stationary regular sequence has a limiting distribution depending only on the spectral density of the sequence. We characterize this limit in terms of Stieltjes transform via a certain simple equation. No rate of convergence to zero of the covariances is imposed, so, the underlying process can exhibit long memory. If the stationary sequence has trivial left sigma field the result holds without any other additional assumptions. This is always true if the entries are functions of i.i.d.

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  • Merlevède, F. & Peligrad, M., 2016. "On the empirical spectral distribution for matrices with long memory and independent rows," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2734-2760.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:9:p:2734-2760
    DOI: 10.1016/j.spa.2016.02.016
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    References listed on IDEAS

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    1. Yin, Y. Q., 1986. "Limiting spectral distribution for a class of random matrices," Journal of Multivariate Analysis, Elsevier, vol. 20(1), pages 50-68, October.
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    4. Silverstein, J. W., 1995. "Strong Convergence of the Empirical Distribution of Eigenvalues of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 55(2), pages 331-339, November.
    5. Banna, Marwa & Merlevède, Florence & Peligrad, Magda, 2015. "On the limiting spectral distribution for a large class of symmetric random matrices with correlated entries," Stochastic Processes and their Applications, Elsevier, vol. 125(7), pages 2700-2726.
    6. Rosenblatt, M., 2009. "A comment on a conjecture of N. Wiener," Statistics & Probability Letters, Elsevier, vol. 79(3), pages 347-348, February.
    7. Guangming Pan & Jiti Gao & Yanrong Yang, 2014. "Testing Independence Among a Large Number of High-Dimensional Random Vectors," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(506), pages 600-612, June.
    8. 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.
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    Cited by:

    1. Pavel Yaskov, 2018. "LLN for Quadratic Forms of Long Memory Time Series and Its Applications in Random Matrix Theory," Journal of Theoretical Probability, Springer, vol. 31(4), pages 2032-2055, December.
    2. 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.
    3. Jamshid Namdari & Debashis Paul & Lili Wang, 2021. "High-Dimensional Linear Models: A Random Matrix Perspective," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(2), pages 645-695, August.
    4. Patrice Abry & B. Cooper Boniece & Gustavo Didier & Herwig Wendt, 2023. "Wavelet eigenvalue regression in high dimensions," Statistical Inference for Stochastic Processes, Springer, vol. 26(1), pages 1-32, April.
    5. A. Lytova, 2018. "Central Limit Theorem for Linear Eigenvalue Statistics for a Tensor Product Version of Sample Covariance Matrices," Journal of Theoretical Probability, Springer, vol. 31(2), pages 1024-1057, June.

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