Advertisement

Total Variation Approximation of Random Orthogonal Matrices by Gaussian Matrices

  • Kathryn StewartEmail author
Article
  • 7 Downloads

Abstract

The topic of this paper is the asymptotic distribution of the entries of random orthogonal matrices distributed according to Haar measure. We examine the total variation distance between the joint distribution of the entries of \(W_n\), the \(p_n \times q_n\) upper-left block of a Haar-distributed matrix, and that of \(p_nq_n\) independent standard Gaussian random variables and show that the total variation distance converges to zero when \(p_nq_n = o(n)\).

Keywords

Random orthogonal matrix Central limit theorem Wishart matrices Moments 

Mathematics Subject Classification (2010)

60F05 60C05 

Notes

Acknowledgements

The results of this paper are part of a Ph.D. thesis written under the direction of Elizabeth Meckes; the author very much thanks her for many helpful conversations.

References

  1. 1.
    Bai, Z.D.: Methodologies in spectral analysis of large-dimensional random matrices, a review. Stat. Sin. 9, 611–677 (1999)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bai, Z.D., Silverstein, J.W.: Spectral Analysis of Large Dimensional Random Matrices. Science Press, Beijing (2010)CrossRefzbMATHGoogle Scholar
  3. 3.
    Borel, E.: Introduction géometrique á quelques théories physiques. Gauthier-Villars, Paris (1906)zbMATHGoogle Scholar
  4. 4.
    Chatterjee, S., Meckes, E.: Multivariate normal approximation using exchangeable pairs. ALEA 4, 257–283 (2008)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Diaconis, P.W., Eaton, M.L., Lauritzen, S.L.: Finite de Finetti theorems in linear models and multivariate analysis. Scand. J. Stat. 19, 289–315 (1992)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Diaconis, P.W., Freedman, D.: A dozen de Finetti-style results in search of a theory. Ann. Inst. H. Poincare Probab. Statist. 23, 397–423 (1987)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Eaton, M.L.: Group Invariance Applications in Statistics. IMS, Hayward (1989)zbMATHGoogle Scholar
  8. 8.
    Feller, W.: An Introduction to Probability Theory and its Applications, vol. I, 3rd edn. Wiley, New York (1968)zbMATHGoogle Scholar
  9. 9.
    Jiang, T.: How many entries of a typical orthogonal matrix can be approximated by independent normals? Ann. Probab. 34, 1497–1529 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jiang, T., Ma, Y.: Distance between random orthogonal matrices and independent normals. (2017). arXiv:1704.05205
  11. 11.
    Jiang, T., Qi, Y.: Limiting distributions of likelihood ratio tests for high-dimensional normal distributions. Scand. J. Stat. 42(4), 988–1009 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Yin, Y.Q., Bai, Z.D., Krishnaiah, P.R.: On the limit of the largest eigenvalue of the large-dimensional sample covariance matrix. Probab. Theory Relat. Fields. 78, 509–521 (1988)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsCase Western Reserve UniversityClevelandUSA

Personalised recommendations