Before revision:  In this paper we find sufficient conditions for the desired convergence by means of the weak convergence method to a class of path-dependent stochastic differential equations  .
After revision:  In this paper we find sufficient conditions for the desired convergence by means of the weak convergence method to a class of path-dependent stochastic differential equations . The main ingredient is to identify the weak convergence of the approximated solution to the underlying path-dependent stochastic differential equation .