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Asymptotic theory for the sample covariance matrix of a heavy-tailed multivariate time series

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  • Davis, Richard A.
  • Mikosch, Thomas
  • Pfaffel, Oliver

Abstract

In this paper we give an asymptotic theory for the eigenvalues of the sample covariance matrix of a multivariate time series. The time series constitutes a linear process across time and between components. The input noise of the linear process has regularly varying tails with index α∈(0,4); in particular, the time series has infinite fourth moment. We derive the limiting behavior for the largest eigenvalues of the sample covariance matrix and show point process convergence of the normalized eigenvalues. The limiting process has an explicit form involving points of a Poisson process and eigenvalues of a non-negative definite matrix. Based on this convergence we derive limit theory for a host of other continuous functionals of the eigenvalues, including the joint convergence of the largest eigenvalues, the joint convergence of the largest eigenvalue and the trace of the sample covariance matrix, and the ratio of the largest eigenvalue to their sum.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:3:p:767-799
    DOI: 10.1016/j.spa.2015.10.001
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. 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.
    2. She, Rui & Ling, Shiqing, 2020. "Inference in heavy-tailed vector error correction models," Journal of Econometrics, Elsevier, vol. 214(2), pages 433-450.
    3. 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.
    4. Daisuke Kurisu & Taisuke Otsu, 2021. "Nonparametric inference for extremal conditional quantiles," STICERD - Econometrics Paper Series 616, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    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.

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