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

Natalie Coston () and Sean O’Rourke ()
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Natalie Coston: University of Colorado at Boulder
Sean O’Rourke: University of Colorado at Boulder

Journal of Theoretical Probability, 2020, vol. 33, issue 3, 1541-1612

Abstract: 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.

Keywords: Random matrices; Linear eigenvalue statistics; Non-Hermitian random matrices; iid random matrices; Product matrices; Primary 15A52; Secondary 60F05 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10959-019-00905-0

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