Skip to main content

Limit Theorems for β-Laguerre and β-Jacobi Ensembles


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.

This is a preview of subscription content, access via your institution.


  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

    MathSciNet  Article  Google Scholar 

  3. Dumitriu I, Edelman A. Matrix models for beta ensembles. J Math Phys, 2002, 43(11): 5830–5847

    MathSciNet  Article  Google Scholar 

  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

    MathSciNet  Article  Google Scholar 

  5. Dumitriu I, Paquette E. Global fluctuations for liner statistics of β-Jacobi ensembles. Random Matrices: Theory Appl, 2012, 1(4): 1250013, 60

    MathSciNet  Article  Google Scholar 

  6. Edelman A, Koev P. Eigenvalue distributions of beta-Wishart matrices. Random Matrices: Theory Appl, 2014, 3(2): 1450009

    MathSciNet  Article  Google Scholar 

  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

    MathSciNet  Article  Google Scholar 

  8. Gerŝgorin S A. Uber die abgrenzung der eigenwerte einer matrix. Nauk SSSR Ser Fiz-Mat, 1931, 6: 749–754

    MATH  Google Scholar 

  9. Jiang T. Limit theorems for beta-Jacobi ensembles. Bernoulli, 2013, 19(3): 1028–1046

    MathSciNet  Article  Google Scholar 

  10. Killip R, Nenciu I. Matrix models for circular ensembles. Int Math Res Not, 2004, 50: 2665–2701

    MathSciNet  Article  Google Scholar 

  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

    MathSciNet  Article  Google Scholar 

  14. Trinh K. On spectral measures of random Jacobi matrices. Osaka J Math, 2018, 55: 595–617

    MathSciNet  MATH  Google Scholar 

  15. Wishart J. The generalized product moment distribution in samples from a normal multivariate population. Biometrika A, 1928, 20: 32–43

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Yutao Ma.

Additional information

Yutao Ma was supported by NSFC (12171038, 11871008).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Huang, N., Ma, Y. Limit Theorems for β-Laguerre and β-Jacobi Ensembles. Acta Math Sci 42, 2025–2039 (2022).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI:

Key words

  • beta-ensembles
  • largest and smallest eigenvalues
  • central limit theorem
  • total variation distance

2010 MR Subject Classification

  • 15B52
  • 60B20
  • 60F10