LIMIT THEOREMS FOR β-LAGUERRE AND β-JACOBI ENSEMBLES

doi:10.1007/s10473-022-0517-x

Acta mathematica scientia,Series B ›› 2022, Vol. 42 ›› Issue (5): 2025-2039.doi: 10.1007/s10473-022-0517-x

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LIMIT THEOREMS FOR β-LAGUERRE AND β-JACOBI ENSEMBLES

Naqi HUANG¹, Yutao MA²

1. Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, Mekelweg 4, 2628 CD, Delft, The Netherlands;
2. School of Mathematical Sciences & Laboratory of Mathematics and Complex Systems of Ministry of Education, Beijing Normal University, Beijing, 100875, China

Received:2020-12-01 Revised:2022-05-20 Published:2022-11-02
Contact: Yutao Ma,E-mail:mayt@bnu.edu.cn E-mail:mayt@bnu.edu.cn
Supported by:
Yutao Ma was supported by NSFC (12171038, 11871008).

Abstract

Abstract: We use tridiagonal models to study the limiting behavior of β-Laguerre and β-Jacobi ensembles, focusing on the limiting behavior of the extremal eigenvalues and the central limit theorem for the two ensembles. For the central limit theorem of β-Laguerre ensembles, we follow the idea in [1] while giving a modified version for the generalized case. Then we use the total variation distance between the two sorts of ensembles to obtain the limiting behavior of β-Jacobi ensembles.

Key words: beta-ensembles, largest and smallest eigenvalues, central limit theorem, total variation distance

CLC Number:

15B52

Naqi HUANG, Yutao MA. LIMIT THEOREMS FOR β-LAGUERRE AND β-JACOBI ENSEMBLES[J].Acta mathematica scientia,Series B, 2022, 42(5): 2025-2039.

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References

[1] Dumitriu I. Eigenvalue Statistics for Beta-Ensembles [D]. Massachusetts Institute of Technology, 2003
[2] Dumitriu I, Edelman A. Global spectrum fluctuations for the β-Hermite and β-Laguerre ensembles via matrix models. J Math Phys, 2006, 47(6): 063302
[3] Dumitriu I, Edelman A. Matrix models for beta ensembles. J Math Phys, 2002, 43(11): 5830–5847
[4] Dumitriu I, Koev P. Distributions of the extreme eigenvalues of beta-Jacobi random matrices. SIAM J Matrix Anal Appl, 2008, 30(1): 1–6
[5] Dumitriu I, Paquette E. Global fluctuations for liner statistics of β-Jacobi ensembles. Random Matrices: Theory Appl, 2012, 1(4): 1250013, 60
[6] Edelman A, Koev P. Eigenvalue distributions of beta-Wishart matrices. Random Matrices: Theory Appl, 2014, 3(2): 1450009
[7] Edelman A, Sutton B D. The beta-Jacobi matrix model, the CS decomposition, and generalized singular value problems. Found Comput Math, 2008, 8: 259–285
[8] Gerŝgorin S A. Uber die abgrenzung der eigenwerte einer matrix. Nauk SSSR Ser Fiz-Mat, 1931, 6: 749–754
[9] Jiang T. Limit theorems for beta-Jacobi ensembles. Bernoulli, 2013, 19(3): 1028–1046
[10] Killip R, Nenciu I. Matrix models for circular ensembles. Int Math Res Not, 2004, 50: 2665–2701
[11] Killip R. Gaussian fluctuations for β ensembles. Int Math Res Not, 2008, 2008: Art rnn007
[12] Ma Y, Shen X. Approximation of beta-Jocobi ensembles by beta-Laguerre ensembles. To appear at Front Math China, 2022
[13] Silverstein J W. The Smallest eigenvalue of a large dimensional Wishart matrix. Ann Probab, 1985, 13: 1364–1368
[14] Trinh K. On spectral measures of random Jacobi matrices. Osaka J Math, 2018, 55: 595–617
[15] Wishart J. The generalized product moment distribution in samples from a normal multivariate population. Biometrika A, 1928, 20: 32–43

LIMIT THEOREMS FOR β-LAGUERRE AND β-JACOBI ENSEMBLES

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