The Annals of Statistics

The complex Wishart distribution and the symmetric group

Piotr Graczyk, Gérard Letac, and Hélène Massam

Full-text: Open access


Let V be the space of (r,r) Hermitian matrices and let $\Omega$ be the cone of the positive definite ones. We say that the random variable S, taking its values in $\overline{\Omega},$ has the complex Wishart distribution $\gamma_{p,\sigma}$ if $\mathbb{E}(\exp \,\tr (\theta S))=(\det (I_r-\sigma\theta))^{-p},$ where $\sigma$ and $\sigma^{-1}-\theta$ are in $\Omega,$ and where p=1,2,...,r-1 or p>r-1. In this paper, we compute all moments of $S$ and $S^{-1}.$ The techniques involve in particular the use of the irreducible characters of the symmetric group.

Article information

Ann. Statist., Volume 31, Number 1 (2003), 287-309.

First available in Project Euclid: 26 February 2003

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H05: Characterization and structure theory
Secondary: 60E05: Distributions: general theory 62E17: Approximations to distributions (nonasymptotic)

Complex Wishart moments symmetric group irreducible representations Schur polynomials


Graczyk, Piotr; Letac, Gérard; Massam, Hélène. The complex Wishart distribution and the symmetric group. Ann. Statist. 31 (2003), no. 1, 287--309. doi:10.1214/aos/1046294466.

Export citation


  • BRILLINGER, D. R. and KRISHNAIAH, P. R., eds. (1983). Time Series in the Frequency Domain. North-Holland, Amsterdam.
  • CAPITAINE, M. and CASALIS, M. (2002). Asy mptotic freeness by generalized moments for Gaussian and Wishart matrices. Application to Beta random matrices. Unpublished manuscript.
  • CASALIS, M. and LETAC, G. (1994). Characterization of the Jorgensen set in the generalized linear model. Test 3 145-162.
  • FULTON, W. (1997). Young Tableaux. Cambridge Univ. Press.
  • FULTON, W. and HARRIS, J. (1991). Representation Theory. Springer, New York.
  • GAP99 (1999). The GAP group, GAP-groups, algorithms, and programming, version 4.2. Aachen, St. Andrews. Available at gap.
  • GOODMAN, N. R. (1963). Statistical analysis based on a certain multivariate complex Gaussian distribution. Ann. Math. Statist. 34 152-177.
  • GRACZy K, P., LETAC, G. and MASSAM, H. (2000). The complex Wishart distribution and the sy mmetric group. Available at
  • JAMES, G. and KERBER, A. (1981). The Representation Theory of the Sy mmetric Group. AddisonWesley, Reading, MA.
  • LETAC, G. and MASSAM, H. (1998). Quadratic and inverse regressions for Wishart distributions. Ann. Statist. 26 573-595.
  • LETAC, G. and MASSAM, H. (2000). Representations of the Wishart distributions. In Probability on Algebraic Structures (G. Budzban, P. Feinsilver and A. Mukherjea, eds.) 121-142. Amer. Math. Soc., Providence, RI.
  • MACDONALD, I. G. (1995). Sy mmetric Functions and Hall Poly nomials, 2nd ed. Clarendon Press, Oxford.
  • MAIWALD, D. and KRAUS, D. (2000). Calculation of moments of complex Wishart and complex inverse Wishart distributed matrices. IEEE Proc. Radar Sonar Navigation 147 162-168.
  • SIMON, B. (1996). Representations of Finite and Compact Groups. Amer. Math. Soc., Providence, RI.
  • STANLEY, R. P. (1971). Theory and application of plane partitions. I, II. Studies in Appl. Math. 50 167-187, 259-279.
  • VON ROSEN, D. (1988). Moments for the inverted Wishart distribution. Scand. J. Statist. 15 97-109.
  • WONG, C. S. and LIU, D. (1995). Moments of generalized Wishart distributions. J. Multivariate Anal. 52 280-294.