Reinforced weak convergence of stochastic processes (Q2483844)

The sequence of stochastic processes \(X_n\) on \(C[0,1]\) is considered. This sequence is polynomially convergent if \(EF(X_n)\to EF(X)\) for continuous functionals \(F\) of polynomial growth. The main theorem gives two conditions for the polynomial convergence \(X_n\) to \(X\). The polynomial convergence is implied by the ``strong approximation''. The main theorem applies to the contour of simple trees, that is the process of the height of leaves. The polynomial convergence of two important classes of processes to the Brownian excursion is proved. Some discrete excursions converge polynomially to the Brownian excursion. The depth first walk associated to simple trees can have the same property.