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

TL;DR

This paper derives explicit formulas for the moments of the inverse of real Wishart matrices using orthogonal Weingarten functions, providing new tools for understanding their statistical properties.

Contribution

It introduces a method to compute general moments of inverse Wishart matrices employing orthogonal Weingarten functions, expanding analytical capabilities.

Findings

Explicit formulas for moments of inverse Wishart matrices.

Formulas for moments of traces of Wishart matrices and their inverses.

Application of orthogonal Weingarten functions to random matrix theory.

Abstract

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.