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Almost sure convergence of the largest and smallest eigenvalues of high-dimensional sample correlation matrices

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  • Heiny, Johannes
  • Mikosch, Thomas

Abstract

In this paper, we show that the largest and smallest eigenvalues of a sample correlation matrix stemming from n independent observations of a p-dimensional time series with iid components converge almost surely to (1+γ)2 and (1−γ)2, respectively, as n→∞, if p∕n→γ∈(0,1] and the truncated variance of the entry distribution is “almost slowly varying”, a condition we describe via moment properties of self-normalized sums. Moreover, the empirical spectral distributions of these sample correlation matrices converge weakly, with probability 1, to the Marčenko–Pastur law, which extends a result in Bai and Zhou (2008). We compare the behavior of the eigenvalues of the sample covariance and sample correlation matrices and argue that the latter seems more robust, in particular in the case of infinite fourth moment. We briefly address some practical issues for the estimation of extreme eigenvalues in a simulation study.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:8:p:2779-2815
    DOI: 10.1016/j.spa.2017.10.002
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    References listed on IDEAS

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    1. 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.
    2. Bai, Z. D. & Silverstein, Jack W. & Yin, Y. Q., 1988. "A note on the largest eigenvalue of a large dimensional sample covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 26(2), pages 166-168, August.
    3. Jonsson, Fredrik, 2010. "On the quadratic moment of self-normalized sums," Statistics & Probability Letters, Elsevier, vol. 80(17-18), pages 1289-1296, September.
    4. 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.
    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.
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    Cited by:

    1. 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.
    2. Gusakova, Anna & Heiny, Johannes & Thäle, Christoph, 2023. "The volume of random simplices from elliptical distributions in high dimension," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 357-382.

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