Reinforced weak convergence of stochastic processes

We consider a sequence of stochastic processes X, on C[0, 1] converging weakly to X and call it polynomially convergent, if EF(X-n) --> EF(X) for continuous functionals F of polynomial growth. We present a sufficient moment conditions on X, 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. (C) 2004 Elsevier B.V. All rights reserved.