Contributions to the Theory of Dirichlet Processes,

The authors derive some basic properties of a sample X1,...,Xn from a Dirichlet process. Let ri 0 if Xi Xk for some k 1, ..., i-1, and 1 otherwise. They authors first establish the distribution of the summation from i1 to n of ri, the number of distinct observations in the sample, and certain conditional and unconditional joint distributions of the Xis and ris. These results are used to prove a weak law of large numbers for Z sub n the summation from i1 to n of ri Xi the summation from i1 to n of ri. The weak law is then applied to obtain the consistency of a Bayes estimator of the index of the transition measure of a mixture of Dirichlet processes. Author