Beta-hypergeometric probability distribution on symmetric matrices

Some remarkable properties of the beta distribution are based on relations involving independence between beta random variables such that a parameter of one among them is the sum of the parameters of an other (see (1.1) et (1.2) below). Asci, Letac and Piccioni \cite{6} have used the real beta-hypergeometric distribution on $ \reel$ to give a general version of these properties without the condition on the parameters. In the present paper, we extend the properties of the real beta to the beta distribution on symmetric matrices, we use on the positive definite matrices the division algorithm defined by the Cholesky decomposition to define a matrix-variate beta-hypergeometric distribution, and we extend to this distribution the proprieties established in the real case by Asci, Letac and Piccioni