Asymptotic normality for traces of polynomials in independent complex Wishart matrices

Probability Theory and Related Fields

Volumn 140, Issue 3-4, 2008, Pages 383-405

  Bryc, Włodzimierz ^a  

^a UNIVERSITY OF CINCINNATI   (United States)

Author keywords

Complex Wishart; Moments; Normal approximation

Indexed keywords

EID: 36749047858     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00440-007-0068-z     Document Type: Article

References (29)

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
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
4. Biane P. (1997). Some properties of crossings and partitions. Discret. Math. 175(1-3): 41-53
5. Cabanal-Duvillard T. (2001). Fluctuations de la loi empirique de grandes matrices aléatoires. Ann. Inst. H. Poincaré Probab. Stat. 37(3): 373-402
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
10. Goodman N.R. (1963). Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction). Ann. Math. Stat. 34: 152-177
11. Graczyk P., Letac G. and Massam H. (2003). The complex Wishart distribution and the symmetric group. Ann. Stat. 31(1): 287-309
12. Guionnet, A., Maurel-Segala, E.: Combinatorial aspects of matrix models. ALEA Lat. Am. J. Probab. Math. Stat. 1, 241-279 (electronic) (2006)
13. Haagerup U. and Thorbjørnsen S. (2003). Random matrices with complex Gaussian entries. Expo. Math. 21(4): 293-337
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
16. Jonsson D. (1982). Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal. 12(1): 1-38
17. Khatri C.G. (1965). Classical statistical analysis based on a certain multivariate complex Gaussian distribution. Ann. Math. Stat. 36: 98-114
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
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
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)
23. Machì A. (1984). The Riemann-Hurwitz formula for the centralizer of a pair of permutations. Arch. Math. (Basel) 42(3): 280-288
24. Marcinkiewicz J. (1939). Sur une propriété de la loi de Gauß. Math. Z. 44(1): 612-618
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
27. Speed T.P. (1983). Cumulants and partition lattices. Aust. J. Stat. 25(2): 378-388
28. Srivastava M.S. (1965). On the complex Wishart distribution. Ann. Math. Stat. 36: 313-315
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