Reinforced weak convergence of stochastic processes

We consider a sequence of stochastic processes Xn on C[0,1] converging weakly to X and call it polynomially convergent, if EF(Xn)→EF(X) for continuous functionals F of polynomial growth. We present a sufficient moment conditions on Xn for polynomial convergence and provide several examples, e.g. discrete excursions and depth first path associated to Galton-Watson trees. This concept leads to a new approach to moments of functionals of rooted trees such as height and path length.