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Joint Central Limit Theorem for Eigenvalue Statistics from Several Dependent Large Dimensional Sample Covariance Matrices with Application

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  • Weiming Li
  • Zeng Li
  • Jianfeng Yao

Abstract

Let Xn = (xij) be a k×n data matrix with complex‐valued, independent and standardized entries satisfying a Lindeberg‐type moment condition. We consider simultaneously R sample covariance matrices Bnr=1nQrXnXn∗Qr⊤,1≤r≤R, where the Qr's are non‐random real matrices with common dimensions p×k(k≥p). Assuming that both the dimension p and the sample size n grow to infinity, the limiting distributions of the eigenvalues of the matrices {Bnr} are identified, and as the main result of the paper, we establish a joint central limit theorem (CLT) for linear spectral statistics of the R matrices {Bnr}. Next, this new CLT is applied to the problem of testing a high‐dimensional white noise in time series modelling. In experiments, the derived test has a controlled size and is significantly faster than the classical permutation test, although it does have lower power. This application highlights the necessity of such joint CLT in the presence of several dependent sample covariance matrices. In contrast, all the existing works on CLT for linear spectral statistics of large sample covariance matrices deal with a single sample covariance matrix (R = 1).

Suggested Citation

  • Weiming Li & Zeng Li & Jianfeng Yao, 2018. "Joint Central Limit Theorem for Eigenvalue Statistics from Several Dependent Large Dimensional Sample Covariance Matrices with Application," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 45(3), pages 699-728, September.
  • Handle: RePEc:bla:scjsta:v:45:y:2018:i:3:p:699-728
    DOI: 10.1111/sjos.12320
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    Cited by:

    1. Wang, Zhendong & Xu, Xingzhong, 2021. "Testing high dimensional covariance matrices via posterior Bayes factor," Journal of Multivariate Analysis, Elsevier, vol. 181(C).
    2. Tingting Zou & Shurong Zheng & Zhidong Bai & Jianfeng Yao & Hongtu Zhu, 2022. "CLT for linear spectral statistics of large dimensional sample covariance matrices with dependent data," Statistical Papers, Springer, vol. 63(2), pages 605-664, April.
    3. Bose, Arup & Hachem, Walid, 2020. "Smallest singular value and limit eigenvalue distribution of a class of non-Hermitian random matrices with statistical application," Journal of Multivariate Analysis, Elsevier, vol. 178(C).

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