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A Functional CLT for Partial Traces of Random Matrices

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  • Jan Nagel

    (TU Dortmund, Fakultät für Mathematik)

Abstract

In this paper, we show a functional central limit theorem for the sum of the first $$\lfloor t n \rfloor $$ ⌊ t n ⌋ diagonal elements of f(Z) as a function in t, for Z a random real symmetric or complex Hermitian $$n\times n$$ n × n matrix. The result holds for orthogonal or unitarily invariant distributions of Z, in the cases when the linear eigenvalue statistic $${\text {tr}}f(Z)$$ tr f ( Z ) satisfies a central limit theorem (CLT). The limit process interpolates between the fluctuations of individual matrix elements as $$f(Z)_{1,1}$$ f ( Z ) 1 , 1 and of the linear eigenvalue statistic. It can also be seen as a functional CLT for processes of randomly weighted measures.

Suggested Citation

  • Jan Nagel, 2021. "A Functional CLT for Partial Traces of Random Matrices," Journal of Theoretical Probability, Springer, vol. 34(2), pages 953-974, June.
    • Handle: RePEc:spr:jotpro:v:34:y:2021:i:2:d:10.1007_s10959-019-00982-1
    DOI: 10.1007/s10959-019-00982-1
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    References listed on IDEAS

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    1. Sean O’Rourke & David Renfrew & Alexander Soshnikov, 2013. "On Fluctuations of Matrix Entries of Regular Functions of Wigner Matrices with Non-identically Distributed Entries," Journal of Theoretical Probability, Springer, vol. 26(3), pages 750-780, September.
    2. Jonsson, Dag, 1982. "Some limit theorems for the eigenvalues of a sample covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 12(1), pages 1-38, March.
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