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Spectral Radii of Products of Random Rectangular Matrices

Author

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  • Yongcheng Qi

    (University of Minnesota Duluth)

  • Mengzi Xie

    (University of Minnesota Duluth)

Abstract

We consider m independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables. Assume the product of the m rectangular matrices is an n-by-n square matrix. The maximum absolute value of the n eigenvalues of the product matrix is called spectral radius. In this paper, we study the limiting spectral radii of the product when m changes with n and can even diverge. We give a complete description for the limiting distribution of the spectral radius. Our results reduce to those in Jiang and Qi (J Theor Probab 30(1):326–364, 2017) when the rectangular matrices are square.

Suggested Citation

  • Yongcheng Qi & Mengzi Xie, 2020. "Spectral Radii of Products of Random Rectangular Matrices," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2185-2212, December.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:4:d:10.1007_s10959-019-00942-9
    DOI: 10.1007/s10959-019-00942-9
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    References listed on IDEAS

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    1. J. P. Bouchaud & M. Potters, 2009. "Financial Applications of Random Matrix Theory: a short review," Papers 0910.1205, arXiv.org.
    2. J.-P. Bouchaud & L. Laloux & M. A. Miceli & M. Potters, 2007. "Large dimension forecasting models and random singular value spectra," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 55(2), pages 201-207, January.
    3. Giacomo Livan & Luca Rebecchi, 2012. "Asymmetric correlation matrices: an analysis of financial data," Papers 1201.6535, arXiv.org, revised Apr 2012.
    4. Zeng, Xingyuan, 2017. "Limiting empirical distribution for eigenvalues of products of random rectangular matrices," Statistics & Probability Letters, Elsevier, vol. 126(C), pages 33-40.
    5. Tiefeng Jiang & Yongcheng Qi, 2017. "Spectral Radii of Large Non-Hermitian Random Matrices," Journal of Theoretical Probability, Springer, vol. 30(1), pages 326-364, March.
    6. Chang, Shuhua & Qi, Yongcheng, 2017. "Empirical distribution of scaled eigenvalues for product of matrices from the spherical ensemble," Statistics & Probability Letters, Elsevier, vol. 128(C), pages 8-13.
    7. Tiefeng Jiang & Yongcheng Qi, 2019. "Empirical Distributions of Eigenvalues of Product Ensembles," Journal of Theoretical Probability, Springer, vol. 32(1), pages 353-394, March.
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