On the convergence of Dirichlet processes

For a given weakly convergent sequence {X-n} of Dirichlet processes we show weak convergence of the. sequence of the corresponding quadratic variation processes as well as stochastic integrals driven by the X-n values provide d that the condition UTD (a counterpart to the condition UT for Dirichlet p rocesses) holds true. Moreover, we show that under UTD the limit process of {X-n} is a Dirichlet process, too.