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 Zh. Mat. Fiz. Anal. Geom., 2013, Volume 9, Number 4, Pages 536–581 (Mi jmag579)

On Non-Gaussian Limiting Laws for Certain Statistics of Wigner Matrices

A. Lytova

B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, 47 Lenin Ave., Kharkiv 61103, Ukraine

Abstract: This paper is a continuation of our papers [12–14] in which the limiting laws of fluctuations were found for the linear eigenvalue statistics $\mathrm{Tr} \varphi (M^{(n)})$ and for the normalized matrix elements $\sqrt{n}\varphi_{jj}(M^{(n)})$ of differentiable functions of real symmetric Wigner matrices $M^{(n)}$ as $n\rightarrow\infty$. Here we consider another spectral characteristic of Wigner matrices, $\xi^{A} _{n}[\varphi ]=\mathrm{Tr} \varphi (M^{(n)})A^{(n)}$, where $\{A^{(n)}\}_{n=1}^\infty$ is a certain sequence of non-random matrices. We show first that if $M^{(n)}$ belongs to the Gaussian Orthogonal Ensemble, then $\xi^{A} _{n}[\varphi ]$ satisfies the Central Limit Theorem. Then we consider Wigner matrices with i.i.d. entries possessing the entire characteristic function and find the limiting probability law for $\xi^{A} _{n}[\varphi ]$, which in general is not Gaussian.

Key words and phrases: Wigner matrices, spectral characteristics, central limit theorem.

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Bibliographic databases:
MSC: Primary 60F05, 15B52; Secondary 15A18
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Citation: A. Lytova, “On Non-Gaussian Limiting Laws for Certain Statistics of Wigner Matrices”, Zh. Mat. Fiz. Anal. Geom., 9:4 (2013), 536–581

Citation in format AMSBIB
\Bibitem{Lyt13} \by A.~Lytova \paper On Non-Gaussian Limiting Laws for Certain Statistics of~Wigner Matrices \jour Zh. Mat. Fiz. Anal. Geom. \yr 2013 \vol 9 \issue 4 \pages 536--581 \mathnet{http://mi.mathnet.ru/jmag579} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3155024} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000325604900006} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. V. Vasilchuk, “On the Fluctuations of Entries of Matrices whose Randomness is due to Classical Groups”, Zhurn. matem. fiz., anal., geom., 10:4 (2014), 451–484
2. A. Lytova, L. Pastur, “On a limiting distribution of singular values of random band matrices”, Zhurn. matem. fiz., anal., geom., 11:4 (2015), 311–332
3. L. Erdos, D. Schroder, “Fluctuations of functions of Wigner matrices”, Electron. Commun. Probab., 21 (2016), 15
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