On the convergence of Dirichlet processes

Abstract

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