Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Asymptotic normality for traces of polynomials in independent complex Wishart matrices
Download PDF
Download PDF
  • Published: 03 April 2007

Asymptotic normality for traces of polynomials in independent complex Wishart matrices

  • Włodzimierz Bryc1 

Probability Theory and Related Fields volume 140, pages 383–405 (2008)Cite this article

  • 112 Accesses

  • 2 Citations

  • Metrics details

Abstract

We derive a non-asymptotic expression for the moments of traces of monomials in several independent complex Wishart matrices, extending some explicit formulas available in the literature. We then deduce the explicit expression for the cumulants. From the latter, we read out the multivariate normal approximation to the traces of finite families of polynomials in independent complex Wishart matrices.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Anderson, T.W.: An introduction to multivariate statistical analysis, third edn. Wiley Series in Probability and Statistics. Wiley-Interscience, Hoboken (2003)

  2. Arharov L.V. (1971). Limit theorems for the characteristic roots of a sample covariance matrix. Dokl. Akad. Nauk SSSR 199: 994–997

    MathSciNet  Google Scholar 

  3. Bai Z.D. and Silverstein J.W. (2004). CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32(1A): 553–605

    Article  MATH  MathSciNet  Google Scholar 

  4. Biane P. (1997). Some properties of crossings and partitions. Discret. Math. 175(1–3): 41–53

    Article  MATH  MathSciNet  Google Scholar 

  5. Cabanal-Duvillard T. (2001). Fluctuations de la loi empirique de grandes matrices aléatoires. Ann. Inst. H. Poincaré Probab. Stat. 37(3): 373–402

    Article  MATH  MathSciNet  Google Scholar 

  6. Collins, B., Mingo, J.A., Sniady, P., Speicher, R.: Second Order Freeness and Fluctuations of Random Matrices, III. Higher order freeness and free cumulants. arxiv (2006). http://arxiv.org.math.OA/0606431. Preprint

  7. Diaconis, P., Shahshahani, M.: On the eigenvalues of random matrices. J. Appl. Probab. 31A, 49–62 (1994). Studies in applied probability

  8. Edelman, A., Mingo, J., Rao, N.R., Speicher, R.: Statistical eigen-inference from large Wishart matrices (2006). http://arxiv.org.math.ST/0701314. Preprint

  9. Goodman N.R. (1963). The distribution of the determinant of a complex Wishart distributed matrix. Ann. Math. Stat. 34: 178–180

    Google Scholar 

  10. Goodman N.R. (1963). Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction). Ann. Math. Stat. 34: 152–177

    Google Scholar 

  11. Graczyk P., Letac G. and Massam H. (2003). The complex Wishart distribution and the symmetric group. Ann. Stat. 31(1): 287–309

    Article  MATH  MathSciNet  Google Scholar 

  12. Guionnet, A., Maurel-Segala, E.: Combinatorial aspects of matrix models. ALEA Lat. Am. J. Probab. Math. Stat. 1, 241–279 (electronic) (2006)

    Google Scholar 

  13. Haagerup U. and Thorbjørnsen S. (2003). Random matrices with complex Gaussian entries. Expo. Math. 21(4): 293–337

    MATH  MathSciNet  Google Scholar 

  14. Hanlon, P.J., Stanley, R.P., Stembridge, J.R.: Some combinatorial aspects of the spectra of normally distributed random matrices. In: Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991), Contemp. Math. vol. 138, pp. 151–174. Am. Math. Soc., Providence, RI (1992)

  15. Jacques A. (1968). Sur le genre d’une paire de substitutions. C. R. Acad. Sci. Paris Sér. A-B 267: A625–A627

    MathSciNet  Google Scholar 

  16. Jonsson D. (1982). Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal. 12(1): 1–38

    Article  MATH  MathSciNet  Google Scholar 

  17. Khatri C.G. (1965). Classical statistical analysis based on a certain multivariate complex Gaussian distribution. Ann. Math. Stat. 36: 98–114

    MathSciNet  Google Scholar 

  18. Khatri C.G. (1965). A test for reality of a covariance matrix in a certain complex Gaussian distribution. Ann. Math. Stat. 36: 115–119

    MathSciNet  Google Scholar 

  19. Kusalik, T., Mingo, J.A., Speicher, R.: Orthogonal Polynomials and Fluctuations of Random Matrices (2005). http://arxiv.org.math.OA/0503169. Preprint

  20. Lando, S.K., Zvonkin, A.K.: Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, vol. 141. Springer, Berlin (2004). With an appendix by Don B. Zagier, Low-Dimensional Topology, II

  21. Leonov V.P. and Shiryaev A.N. (1959). On a method of semi-invariants. Theor. Probab. Appl. 4: 319–329

    Article  Google Scholar 

  22. Lu, I.L., Richards, D.S.P.: MacMahon’s master theorem, representation theory, and moments of Wishart distributions. Adv. Appl. Math. 27(2–3), 531–547 (2001). Special issue in honor of Dominique Foata’s 65th birthday (Philadelphia, PA, 2000)

    Google Scholar 

  23. Machì A. (1984). The Riemann–Hurwitz formula for the centralizer of a pair of permutations. Arch. Math. (Basel) 42(3): 280–288

    MATH  MathSciNet  Google Scholar 

  24. Marcinkiewicz J. (1939). Sur une propriété de la loi de Gauß. Math. Z. 44(1): 612–618

    Article  MathSciNet  Google Scholar 

  25. Maurel-Segala, E.: High order expansion of matrix models and enumeration of maps (2006). ArXiv:math.PR/0608192

  26. Mingo J.A. and Nica A. (2004). Annular noncrossing permutations and partitions and second-order asymptotics for random matrices. Int. Math. Res. Notes 28: 1413–1460

    Article  MathSciNet  Google Scholar 

  27. Speed T.P. (1983). Cumulants and partition lattices. Aust. J. Stat. 25(2): 378–388

    MATH  MathSciNet  Google Scholar 

  28. Srivastava M.S. (1965). On the complex Wishart distribution. Ann. Math. Stat. 36: 313–315

    Google Scholar 

  29. Stanley, R.P.: Enumerative combinatorics. vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62. Cambridge University Press, Cambridge (1999). With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Cincinnati, 2855 Campus Way, PO Box 210025, Cincinnati, OH, 45221-0025, USA

    Włodzimierz Bryc

Authors
  1. Włodzimierz Bryc
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Włodzimierz Bryc.

Additional information

Research partially supported by NSF grant #DMS-0504198.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bryc, W. Asymptotic normality for traces of polynomials in independent complex Wishart matrices. Probab. Theory Relat. Fields 140, 383–405 (2008). https://doi.org/10.1007/s00440-007-0068-z

Download citation

  • Received: 26 October 2006

  • Revised: 11 January 2007

  • Published: 03 April 2007

  • Issue Date: March 2008

  • DOI: https://doi.org/10.1007/s00440-007-0068-z

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Complex Wishart
  • Moments
  • Normal approximation

Mathematics Subject Classification (2000)

  • 62H05
  • 15A52
  • 05C30
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature