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A note on a Marčenko–Pastur type theorem for time series

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  • Yao, Jianfeng

Abstract

In this note we develop an extension of the Marčenko–Pastur theorem to time series model with temporal correlations. The limiting spectral distribution (LSD) of the sample covariance matrix is characterised by an explicit equation for its Stieltjes transform depending on the spectral density of the time series. A numerical algorithm is then given to compute the density functions of these LSD’s.

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  • Yao, Jianfeng, 2012. "A note on a Marčenko–Pastur type theorem for time series," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 22-28.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:1:p:22-28
    DOI: 10.1016/j.spl.2011.08.011
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    References listed on IDEAS

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    1. Jin, Baisuo & Wang, Cheng & Miao, Baiqi & Lo Huang, Mong-Na, 2009. "Limiting spectral distribution of large-dimensional sample covariance matrices generated by VARMA," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 2112-2125, October.
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    3. Jonsson, Dag, 1982. "Some limit theorems for the eigenvalues of a sample covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 12(1), pages 1-38, March.
    4. 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.
    5. Silverstein, J. W. & Choi, S. I., 1995. "Analysis of the Limiting Spectral Distribution of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 54(2), pages 295-309, August.
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    Cited by:

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    3. 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.
    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. Monika Bhattacharjee & Arup Bose, 2017. "Matrix polynomial generalizations of the sample variance-covariance matrix when pn−1 → y ∈ (0, ∞)," Indian Journal of Pure and Applied Mathematics, Springer, vol. 48(4), pages 575-607, December.

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