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Gaussian Fluctuations for Linear Eigenvalue Statistics of Products of Independent iid Random Matrices

Author

Listed:
  • Natalie Coston

    (University of Colorado at Boulder)

  • Sean O’Rourke

    (University of Colorado at Boulder)

Abstract

Consider the product $$X = X_{1}\cdots X_{m}$$ X = X 1 ⋯ X m of m independent $$n\times n$$ n × n iid random matrices. When m is fixed and the dimension n tends to infinity, we prove Gaussian limits for the centered linear spectral statistics of X for analytic test functions. We show that the limiting variance is universal in the sense that it does not depend on m (the number of factor matrices) or on the distribution of the entries of the matrices. The main result generalizes and improves upon previous limit statements for the linear spectral statistics of a single iid matrix by Rider and Silverstein as well as Renfrew and the second author.

Suggested Citation

  • Natalie Coston & Sean O’Rourke, 2020. "Gaussian Fluctuations for Linear Eigenvalue Statistics of Products of Independent iid Random Matrices," Journal of Theoretical Probability, Springer, vol. 33(3), pages 1541-1612, September.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:3:d:10.1007_s10959-019-00905-0
    DOI: 10.1007/s10959-019-00905-0
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    References listed on IDEAS

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    1. Sean O’Rourke & David Renfrew, 2016. "Central Limit Theorem for Linear Eigenvalue Statistics of Elliptic Random Matrices," Journal of Theoretical Probability, Springer, vol. 29(3), pages 1121-1191, September.
    2. Pan, Guangming & Zhou, Wang, 2010. "Circular law, extreme singular values and potential theory," Journal of Multivariate Analysis, Elsevier, vol. 101(3), pages 645-656, March.
    3. Yuriy Nemish, 2018. "No Outliers in the Spectrum of the Product of Independent Non-Hermitian Random Matrices with Independent Entries," Journal of Theoretical Probability, Springer, vol. 31(1), pages 402-444, March.
    4. Edelman, Alan, 1997. "The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law," Journal of Multivariate Analysis, Elsevier, vol. 60(2), pages 203-232, February.
    Full references (including those not matched with items on IDEAS)

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