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Limit theory for the largest eigenvalues of sample covariance matrices with heavy-tails

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  • Davis, Richard A.
  • Pfaffel, Oliver
  • Stelzer, Robert

Abstract

We study the joint limit distribution of the k largest eigenvalues of a p×p sample covariance matrix XXT based on a large p×n matrix X. The rows of X are given by independent copies of a linear process, Xit=∑jcjZi,t−j, with regularly varying noise (Zit) with tail index α∈(0,4). It is shown that a point process based on the eigenvalues of XXT converges, as n→∞ and p→∞ at a suitable rate, in distribution to a Poisson point process with an intensity measure depending on α and ∑cj2. This result is extended to random coefficient models where the coefficients of the linear processes (Xit) are given by cj(θi), for some ergodic sequence (θi), and thus vary in each row of X. As a by-product of our techniques we obtain a proof of the corresponding result for matrices with iid entries in cases where p/n goes to zero or infinity and α∈(0,2).

Suggested Citation

  • Davis, Richard A. & Pfaffel, Oliver & Stelzer, Robert, 2014. "Limit theory for the largest eigenvalues of sample covariance matrices with heavy-tails," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 18-50.
    • Handle: RePEc:eee:spapps:v:124:y:2014:i:1:p:18-50
    DOI: 10.1016/j.spa.2013.07.005
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    References listed on IDEAS

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    1. 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.
    2. Yao, Jianfeng, 2012. "A note on a Marčenko–Pastur type theorem for time series," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 22-28.
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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. Asma Teimouri & Mahbanoo Tata & Mohsen Rezapour & Rafal Kulik & Narayanaswamy Balakrishnan, 2021. "Asymptotic Behavior of Eigenvalues of Variance-Covariance Matrix of a High-Dimensional Heavy-Tailed Lévy Process," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1353-1375, December.
    3. Davis, Richard A. & Mikosch, Thomas & Pfaffel, Oliver, 2016. "Asymptotic theory for the sample covariance matrix of a heavy-tailed multivariate time series," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 767-799.
    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. 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.
    6. Heiny, Johannes & Mikosch, Thomas, 2017. "Eigenvalues and eigenvectors of heavy-tailed sample covariance matrices with general growth rates: The iid case," Stochastic Processes and their Applications, Elsevier, vol. 127(7), pages 2179-2207.

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