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Berry-Esseen bounds and moderate deviations for the norm, entries and spectral radius of products of positive random matrices

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Abstract

Let (gn)n⩾l be a sequence of independent and identically distributed positive random d × d matrices and consider the matrix product Gn = gn⋯ g1. Under suitable conditions, we establish the Berry-Esseen bounds on the rate of convergence in the central limit theorem and Cramér-type moderate deviation expansions, for any matrix norm ∥Gn∥ of Gn, its entries \(G_{n}^{i,j}\) and its spectral radius ρ(Gn). Extended versions of their joint law with the direction of the random walk Gnx are also established, where x is a starting point in the unit sphere of ℝd.

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Acknowledgements

This work was supported by Deutsche Forschungsgemeinschaft (DFG) (Grant No. ME 4473/2-1), the Centre Henri Lebesgue (CHL) (Grant No. ANR-11-LABX-0020-01) and National Natural Science Foundation of China (Grants Nos. 11971063, 11731012, 12271062 and 12288201).

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Authors and Affiliations

  1. Institut für Mathematik und Angewandte Informatik, Universität Hildesheim, Hildesheim, 31141, Germany

    Hui Xiao

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Hui Xiao

  3. LMBA, Université Bretagne Sud, CNRS UMR 6205, Vannes, F-56000, France

    Ion Grama & Quansheng Liu

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  1. Hui Xiao
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  2. Ion Grama
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  3. Quansheng Liu
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Correspondence to Hui Xiao.

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Xiao, H., Grama, I. & Liu, Q. Berry-Esseen bounds and moderate deviations for the norm, entries and spectral radius of products of positive random matrices. Sci. China Math. 67, 627–646 (2024). https://doi.org/10.1007/s11425-021-2067-4

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  • DOI: https://doi.org/10.1007/s11425-021-2067-4

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