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On singular value distribution of large-dimensional autocovariance matrices

Author

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  • Li, Zeng
  • Pan, Guangming
  • Yao, Jianfeng

Abstract

Let (εj)j≥0 be a sequence of independent p-dimensional random vectors and τ≥1 a given integer. From a sample ε1,…,εT+τ of the sequence, the so-called lag-τ auto-covariance matrix is Cτ=T−1∑j=1Tετ+jεjt. When the dimension p is large compared to the sample size T, this paper establishes the limit of the singular value distribution of Cτ assuming that p and T grow to infinity proportionally and the sequence has uniformly bounded (4+δ)th order moments. Compared to existing asymptotic results on sample covariance matrices developed in random matrix theory, the case of an auto-covariance matrix is much more involved due to the fact that the summands are dependent and the matrix Cτ is not symmetric. Several new techniques are introduced for the derivation of the main theorem.

Suggested Citation

  • Li, Zeng & Pan, Guangming & Yao, Jianfeng, 2015. "On singular value distribution of large-dimensional autocovariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 137(C), pages 119-140.
    • Handle: RePEc:eee:jmvana:v:137:y:2015:i:c:p:119-140
    DOI: 10.1016/j.jmva.2015.02.006
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    References listed on IDEAS

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    1. 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.
    2. Lam, Clifford & Yao, Qiwei, 2012. "Factor modeling for high-dimensional time series: inference for the number of factors," LSE Research Online Documents on Economics 45684, London School of Economics and Political Science, LSE Library.
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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. 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.

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