The Matrix-variate Dirichlet Averages and Its Applications

Abstract

This paper is about Dirichlet averages in the matrix-variate case or averages of functions over the Dirichlet measure in the complex domain. The classical power mean contains the harmonic mean, arithmetic mean and geometric mean (Hardy, Littlewood and Polya), which is generalized to $y$-mean by deFinetti and hypergeometric mean by Carlson, see the references herein. Carlson's hypergeometric mean is to average a scalar function over a real scalar variable type-1 Dirichlet measure and this in the current literature is known as Dirichlet average of that function. The idea is examined when there is a type-1 or type-2 Dirichlet density in the complex domain. Averages of several functions are computed in such Dirichlet densities in the complex domain. Dirichlet measures are defined when the matrices are Hermitian positive definite. Some applications are also discussed.