  Motivated by a theorem of Barbour, we revisit some of the classical limit
theorems in probability from the viewpoint of the Stein method. We setup the
framework to bound Wasserstein distances between some distributions on infinite
dimensional spaces. We show that the convergence rate for the Poisson
approximation of the Brownian motion is as expected proportional to
$\lambda^{-1/2}$ where $\lambda$ is the intensity of the Poisson process. We
also exhibit the speed of convergence for the Donsker Theorem and for the
linear interpolation of the Brownian motion. By iterating the procedure, we
give Edgeworth expansions with precise error bounds.
