The hyper Dirichlet process and its discrete approximations: the single conditional independence model

The aim of this paper is the study of some laws of random probability distributions, called hyper Dirichlet processes, charging the product of three sample spaces, with the property that the first and the third components are independent conditional to the second one. The law of the marginals on the first two and on the last two components are specified to be Dirichlet processes with the same marginal parameter measure on the common second component. The joint law is then obtained as the hyper Markov combination of these two laws, as introduced in [3], and in fact these laws are generalizations of the hyper Dirichlet laws on contingency tables considered in the above paper. Our main result is the convergence to the law of a hyper Dirichlet process of the sequence of hyper Dirichlet laws associated to finer and finer ”discretizations” of the two parameter measures, which is proved by means of a suitable coupling construction. MSC: 60G57; 62G99; 62F15; 60F99.