CENTRAL LIMIT THEOREM FOR LINEAR EIGENVALUE STATISTICS OF RANDOM MATRICES WITH INDEPENDENT ENTRIES

@article{Lytova2009CENTRALLT,
  title={CENTRAL LIMIT THEOREM FOR LINEAR EIGENVALUE STATISTICS OF RANDOM MATRICES WITH INDEPENDENT ENTRIES},
  author={Anna Lytova and Leonid A. Pastur},
  journal={Annals of Probability},
  year={2009},
  volume={37},
  pages={1778-1840}
}
We consider n x n real symmetric and Hermitian Wigner random matrices n ―1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n ―1 X*X with independent entries of m x n matrix X. Assuming first that the 4th cumulant (excess) κ 4 of entries of W and X is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as n → ∞, m → ∞, m/n… 

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TLDR
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