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Singular value distribution of dense random matrices with block Markovian dependence

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  • Sanders, Jaron
  • Van Werde, Alexander

Abstract

A block Markov chain is a Markov chain whose state space can be partitioned into a finite number of clusters such that the transition probabilities only depend on the clusters. Block Markov chains thus serve as a model for Markov chains with communities. This paper establishes limiting laws for the singular value distributions of the empirical transition matrix and empirical frequency matrix associated to a sample path of the block Markov chain whenever the length of the sample path is Θ(n2) with n the size of the state space.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:spapps:v:158:y:2023:i:c:p:453-504
    DOI: 10.1016/j.spa.2023.01.001
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    References listed on IDEAS

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    1. Marwa Banna & Florence Merlevède, 2015. "Limiting Spectral Distribution of Large Sample Covariance Matrices Associated with a Class of Stationary Processes," Journal of Theoretical Probability, Springer, vol. 28(2), pages 745-783, June.
    2. 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.
    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).
    4. 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.
    5. Fleermann, Michael & Kirsch, Werner & Kriecherbauer, Thomas, 2021. "The almost sure semicircle law for random band matrices with dependent entries," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 172-200.
    6. Yao, Jianfeng, 2012. "A note on a Marčenko–Pastur type theorem for time series," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 22-28.
    7. Merlevède, F. & Peligrad, M., 2016. "On the empirical spectral distribution for matrices with long memory and independent rows," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2734-2760.
    8. Ding, Xue, 2015. "On some spectral properties of large block Laplacian random matrices," Statistics & Probability Letters, Elsevier, vol. 99(C), pages 61-69.
    9. Olga Friesen & Matthias Löwe, 2013. "The Semicircle Law for Matrices with Independent Diagonals," Journal of Theoretical Probability, Springer, vol. 26(4), pages 1084-1096, December.
    10. Winfried Hochstättler & Werner Kirsch & Simone Warzel, 2016. "Semicircle Law for a Matrix Ensemble with Dependent Entries," Journal of Theoretical Probability, Springer, vol. 29(3), pages 1047-1068, September.
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