Abstract
We study a random positive definite symmetric matrix distributed according to a real Wishart distribution. We compute general moments of the random matrix and of its inverse explicitly. To do so, we employ the orthogonal Weingarten function, which was recently introduced in the study of Haar-distributed orthogonal matrices. As applications, we give formulas for moments of traces of a Wishart matrix and its inverse.
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Collins, B.: Moments and cumulants of polynomial random variables on unitary groups, the Itzykson–Zuber integral, and free probability. Int. Math. Res. Not. 17, 953–982 (2003)
Collins, B., Matsumoto, S.: On some properties of orthogonal Weingarten functions. J. Math. Phys. 50, 113516 (2009)
Collins, B., Śniady, P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Commun. Math. Phys. 264(3), 773–795 (2006)
Graczyk, P., Letac, G., Massam, H.: The complex Wishart distribution and the symmetric group. Ann. Stat. 31(1), 287–309 (2003)
Graczyk, P., Letac, G., Massam, H.: The hyperoctahedral groups, symmetric group representations and the moments of the real Wishart distribution. J. Theor. Probab. 18(1), 1–42 (2005)
Kuriki, S., Numata, Y.: Graph presentations for moments of noncentral Wishart distributions and their applications. Ann. Inst. Stat. Math. 62(4), 645–672 (2010)
Kuriki, S., Numata, Y.: On formulas for moments of the Wishart distributions as weighted generating functions of matchings. Extended Abstract of FPSAC 2010, available at http://math.sfsu.edu/fpsac/pdfpapers/dmAN0172.pdf
Letac, G., Massam, H.: All invariant moments of the Wishart distribution. Scand. J. Stat. 31(2), 295–318 (2004)
Letac, G., Massam, H.: The noncentral Wishart as an exponential family and its moments. J. Multivar. Anal. 99, 1393–1417 (2008)
Lu, I.-L., Richards, D.St.P.: MacMahon’s master theorem, representation theory, and moments of Wishart distributions. Adv. Appl. Math. 27(2–3), 531–547 (2001)
Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford (1995)
Matsumoto, S.: α-Pfaffians, pfaffian point process and shifted Schur measure. Linear Algebra Appl. 403, 369–398 (2005)
Matsumoto, S.: Jucys–Murphy elements, orthogonal matrix integrals, and Jack measures. Preprint, arXiv:1001.2345v1, 35 pp
Matsumoto, S.: General moments of the inverse real Wishart distribution and orthogonal Weingarten functions (preliminary version), available at: http://arxiv.org/abs/1004.4717v2
Matsumoto, S., Novak, J.: Jucys–Murphy elements and unitary matrix integrals. Preprint, arXiv:0905.1992v2, 44 pp
Muirhead, R.J.: Aspects of Multivariate Statistical Theory. Wiley, New York (1982)
Shirai, T.: Remarks on the positivity of α-determinants. Kyushu J. Math. 61, 169–189 (2007)
Vere-Jones, D.: A generalization of permanents and determinants. Linear Algebra Appl. 111, 119–124 (1988)
von Rosen, D.: Moments for the inverted Wishart distribution. Scand. J. Stat. 15(2), 97–109 (1988)
Zinn-Justin, P.: Jucys–Murphy elements and Weingarten matrices. Lett. Math. Phys. 91, 119–127 (2010)
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Matsumoto, S. General Moments of the Inverse Real Wishart Distribution and Orthogonal Weingarten Functions. J Theor Probab 25, 798–822 (2012). https://doi.org/10.1007/s10959-011-0340-0
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DOI: https://doi.org/10.1007/s10959-011-0340-0