On Complex Matrix-Variate Dirichlet Averages and Its Applications in Various Sub-Domains

Area of Science:

• Mathematics
• Probability Theory
• Complex Analysis

Background:

• Classical power means (harmonic, arithmetic, geometric) are well-established.
• De Finetti's y-mean and Carlson's hypergeometric mean generalize these classical means.
• Carlson's mean averages scalar functions over real Dirichlet measures (Dirichlet averages).

Purpose of the Study:

• To extend the concept of Dirichlet averages to the matrix-variate case within the complex domain.
• To investigate Dirichlet densities of Type-1 and Type-2 in the complex plane.
• To compute averages of various functions over these complex Dirichlet densities.

Main Methods:

• Generalization of scalar Dirichlet averages to matrix-variate settings.
• Definition of Dirichlet measures for Hermitian positive definite matrices.
• Computation of function averages over complex Type-1 and Type-2 Dirichlet densities.

Main Results:

• Successful extension of Dirichlet averages to matrix-variate functions in the complex domain.
• Computation of specific function averages over defined complex Dirichlet densities.
• Establishment of Dirichlet measures for Hermitian positive definite matrices.

Conclusions:

• The study provides a theoretical framework for matrix-variate Dirichlet averages in the complex domain.
• The computed averages and defined measures offer new tools for statistical and mathematical analysis.
• Potential applications in various fields are highlighted, warranting further investigation.