Geodesic
On Non-Gaussian Limiting Laws for Certain Statistics of~Wigner Matrices
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 9 (2013), pp. 536-581.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{JMAG_2013_9_a5,
     author = {A. Lytova},
     title = {On {Non-Gaussian} {Limiting} {Laws} for {Certain} {Statistics} {of~Wigner} {Matrices}},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {536--581},
     publisher = {mathdoc},
     volume = {9},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2013_9_a5/}
}
TY  - JOUR
AU  - A. Lytova
TI  - On Non-Gaussian Limiting Laws for Certain Statistics of~Wigner Matrices
JO  - Žurnal matematičeskoj fiziki, analiza, geometrii
PY  - 2013
SP  - 536
EP  - 581
VL  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JMAG_2013_9_a5/
LA  - en
ID  - JMAG_2013_9_a5
ER  - 
%0 Journal Article
%A A. Lytova
%T On Non-Gaussian Limiting Laws for Certain Statistics of~Wigner Matrices
%J Žurnal matematičeskoj fiziki, analiza, geometrii
%D 2013
%P 536-581
%V 9
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JMAG_2013_9_a5/
%G en
%F JMAG_2013_9_a5
A. Lytova. On Non-Gaussian Limiting Laws for Certain Statistics of~Wigner Matrices. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 9 (2013), pp. 536-581. http://geodesic.mathdoc.fr/item/JMAG_2013_9_a5/

[1] G. W. Anderson, O. Zeitouni, “CLT for a Band Matrix Model”, Probab. Theory Related Fields, 134 (2006), 283–338 | DOI | MR | Zbl

[2] Z. D. Bai, B. Q. Miao, G. M. Pan, “On Asymptotics of Eigenvectors of Large Sample Covariance Matrix”, Ann. Probab, 35 (2007), 1532–1572 | DOI | MR | Zbl

[3] Z. D. Bai, J. W. Silverstein, “CLT for Linear Sspectral Statistics of Large Dimensional Sample Covariance Matrices”, Ann. Probab., 32 (2004), 553–605 | DOI | MR | Zbl

[4] V. I. Bogachev, Gaussian measures, Mathematical Surveys and Monographs, 62, Amer. Math. Soc., Providence, RI | MR | Zbl

[5] O. Costin, J. L. Lebowitz, “Gaussian Fluctuations in Random Matrices”, Phys. Rev. Lett., 75 (1995), 69–72 | DOI

[6] L. Erdös, “Universality of Wigner Random Matrices: a Survey of Recent Results”, Uspekhi Mat. Nauk, 66:3(399) (2011), 67–198 | DOI | MR | Zbl

[7] A. Guionnet, “Large Deviations, Upper Bounds, and Central Limit Theorems for Non-commutative Functionals of Gaussian Large Random Matrices”, Ann. Inst. H. Poincaré Probab. Statist., 38 (2002), 341–384 | DOI | MR | Zbl

[8] K. Johansson, “On Fluctuations of Eigenvalues of Random Hermitian Matrices”, Duke Math. J., 91 (1998), 151–204 | DOI | MR | Zbl

[9] B. Khoruzhenko, A. Khorunzhy, L. Pastur, “Asymptotic Properties of Large Random Matrices with Independent Entries”, J. Physics A: Math. and General, 28 (1995), L31–L35 | DOI | MR | Zbl

[10] O. Ledoit, S. Pećhe, Eigenvectors of Some Large Sample Covariance Matrix Ensembles, arXiv: 0911.3010 | MR

[11] A. Lytova, L. Pastur, “Central Limit Theorem for Linear Eigenvalue Statistics of the Wigner and the Sample Covariance Random Matrices”, Metrika, 69:2 (2009), 153–172 | DOI | MR

[12] A. Lytova, L. Pastur, “Central Limit Theorem for Linear Eigenvalue Statistics of Random Matrices with Independent Entries”, Annals Probability, 37:5 (2009), 1778-1840 | DOI | MR | Zbl

[13] A. Lytova, L. Pastur, “Fluctuations of Matrix Elements of Regular Functions of Gaussian Random Matrices”, J. Stat. Phys., 134 (2009), 147–159 | DOI | MR | Zbl

[14] A. Lytova, L. Pastur, Non-Gaussian Limiting Laws for the Entries of Regular Functions of the Wigner Matrices, arXiv: 1103.2345v2

[15] L. Pastur, “A Simple Approach to the Global Regime of Gaussian Ensembles of Random Matrices”, Ukrainian Math. J., 57 (2005), 936–966 | DOI | MR | Zbl

[16] L. Pastur, “Limiting Laws of Linear Eigenvalue Statistics for Unitary Invariant Matrix Models”, J. Math. Phys., 47 (2006), 103303 | DOI | MR | Zbl

[17] L. Pastur, M. Shcherbina, Eigenvalue Distribution of Large Random Matrices, Math. Surv. and Monographs. Amer. Math. Soc., 171, 2011, 634 pp. | DOI | MR | Zbl

[18] A. Pizzo, D. Renfrew, A. Soshnikov, “Fluctuations of Matrix Entries of Regular Functions of Wigner Matrices”, J. Stat. Phys., 146 (2012), 550–591 | DOI | MR | Zbl

[19] S. O'Rourke, D. Renfrew, A. Soshnikov, On Fluctuations of Matrix Entries of Regular Functions of Wigner Matrices with Non-Identically Distributed Entries, arXiv: 1104.1663v4

[20] M. Shcherbina, “Central Limit Theorem for Linear Eigenvalue Statistics of Wigner and Sample Covariance Random Matrices”, J. Math. Phys., Anal., Geom., 7:2 (2011), 176–192 | MR | Zbl

[21] Ya. Sinai, A. Soshnikov, “Central Limit Theorem for Traces of Large Random Symmetric Matrices with Independent Matrix Elements”, Bol. Soc. Brasil. Mat. (N.S.), 29 (1998), 1–24 | DOI | MR | Zbl

[22] A. Soshnikov, “Central Limit Theorem for Local Linear Statistics in Classical Compact Groups and Related Combinatorial Identities”, Ann. Probab., 28 (2000), 1353–1370 | DOI | MR | Zbl