On the Dirichlet-Hypergeometric Distribution on Symmetric Matrices

Abstract

In this paper, we introduce an extension of the real Dirichlet-hypergeometric distribution to symmetric matrices, motivated by the need for structured matrix-valued distributions in multivariate analysis and random matrix theory. This matrix-variate generalization preserves the combinatorial and probabilistic foundations of the classical Dirichlet model while adapting them to the geometric and algebraic structure of symmetric matrices. We study distributional properties, including marginal and conditional distributions, the computation of moments, and derive the distribution of partial matrix sums. Furthermore, we introduce a class of related matrix distributions that naturally emerge in this framework. These results lay a foundation for further theoretical development and potential applications in areas requiring matrix-valued priors or structured random matrices.

Keywords: Dirichlet distribution; Gauss hypergeometric function; Liouville distribution; Transformation; symmetric matrices.

2010 Mathematics Subject Classification. 62E15; 60E05