Finiteness of inverse moments of (m,n,β)-Laguerre matrices

Abstract

Wishart matrices are one of the fundamental matrix models in multivariate statistics. The classical Wishart ensemble has been generalized to [Formula: see text]-Laguerre ensemble for the values of [Formula: see text]. Many properties such as eigenvalue densities, moments and inverse moments of [Formula: see text]-Laguerre matrices are important in various fields of mathematics and physics. The moments and inverse moments of Wishart matrices have been studied rigorously and the explicit formulas were given in [P. Graczyk, G. Letac and H. Massam, Ann. Statist. 31 (2003) 287–309; S. Matsumoto, General moments of the inverse real Wishart distribution and orthogonal Weingarten functions, J. Theoret. Probab. 25 (2012) 798–822]. We give a necessary and sufficient condition for the existence of finite inverse moments of the [Formula: see text]-Laguerre matrix. Moreover, we extend our result for inverse compound Wishart matrices for the values of [Formula: see text] and [Formula: see text].