General moments of the inverse real Wishart distribution and orthogonal Weingarten functions

Details General moments of the inverse real Wishart distribution and orthogonal Weingarten functions General moments of the inverse real Wishart distribution and orthogonal Weingarten functions

2010-04-27

arxiv.org/abs/1004.4717v3

Mathematics

Statistics Theory

29 pages. The last version differs from the published version, but it includes Appendix

Scientific paper

10.1007/s10959-011-0340-0

Let $W$ be a random positive definite symmetric matrix distributed according to a real Wishart distribution and let $W^{-1}=(W^{ij})_{i,j}$ be its inverse matrix. We compute general moments $\mathbb{E} [W^{k_1 k_2} W^{k_3 k_4} ... W^{k_{2n-1}k_{2n}}]$ explicitly. To do so, we employ the orthogonal Weingarten function, which was recently introduced in the study for Haar-distributed orthogonal matrices. As applications, we give formulas for moments of traces of a Wishart matrix and its inverse.

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