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On the limiting spectral distribution for a large class of symmetric random matrices with correlated entries

Author

Listed:
  • Banna, Marwa
  • Merlevède, Florence
  • Peligrad, Magda

Abstract

For symmetric random matrices with correlated entries, which are functions of independent random variables, we show that the asymptotic behavior of the empirical eigenvalue distribution can be obtained by analyzing a Gaussian matrix with the same covariance structure. This class contains both cases of short and long range dependent random fields. The technique is based on a blend of blocking procedure and Lindeberg’s method. This method leads to a variety of interesting asymptotic results for matrices with dependent entries, including applications to linear processes as well as nonlinear Volterra-type processes entries.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:7:p:2700-2726
    DOI: 10.1016/j.spa.2015.01.010
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    Citations

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    Cited by:

    1. 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.
    2. 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.
    3. Fleermann, Michael & Kirsch, Werner & Kriecherbauer, Thomas, 2021. "The almost sure semicircle law for random band matrices with dependent entries," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 172-200.
    4. Heiny, Johannes & Mikosch, Thomas, 2021. "Large sample autocovariance matrices of linear processes with heavy tails," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 344-375.
    5. Alfredas Račkauskas & Charles Suquet, 2023. "Asymptotic Normality in Banach Spaces via Lindeberg Method," Journal of Theoretical Probability, Springer, vol. 36(1), pages 409-455, March.
    6. Heiny, Johannes & Mikosch, Thomas, 2018. "Almost sure convergence of the largest and smallest eigenvalues of high-dimensional sample correlation matrices," Stochastic Processes and their Applications, Elsevier, vol. 128(8), pages 2779-2815.

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