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Zhurnal Matematicheskoi Fiziki, Analiza, Geometrii [Journal of Mathematical Physics, Analysis, Geometry], 2013, Volume 9, Number 4, Pages 536–581
(Mi jmag579)
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This article is cited in 3 scientific papers (total in 3 papers)
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.
Received: 02.04.2012
Citation:
A. Lytova, “On Non-Gaussian Limiting Laws for Certain Statistics of Wigner Matrices”, Zh. Mat. Fiz. Anal. Geom., 9:4 (2013), 536–581
Linking options:
https://www.mathnet.ru/eng/jmag579 https://www.mathnet.ru/eng/jmag/v9/i4/p536
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Abstract page: | 266 | Full-text PDF : | 62 | References: | 47 |
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