## The Annals of Statistics

### The complex Wishart distribution and the symmetric group

#### Abstract

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

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

Dates
First available in Project Euclid: 26 February 2003

https://projecteuclid.org/euclid.aos/1046294466

Digital Object Identifier
doi:10.1214/aos/1046294466

Mathematical Reviews number (MathSciNet)
MR1962508

Zentralblatt MATH identifier
1019.62047

#### Citation

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. https://projecteuclid.org/euclid.aos/1046294466

#### References

• 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 Unpublished manuscript.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 www-gap.dcs.st-and.ac.uk/ 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 www.lsp.ups-tlse.fr/Fp/Letac/index.html.
• 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.
• TORONTO, ONTARIO M3J 1P3 CANADA E-MAIL: massamh@yorku.ca