  According to a version of Donsker's theorem, geodesic random walks on
Riemannian manifolds converge to the respective Brownian motion. From a
computational perspective, however, evaluating geodesics can be quite costly.
We therefore introduce approximate geodesic random walks based on the concept
of retractions. We show that these approximate walks converge in distribution
to the correct Brownian motion as long as the geodesic equation is approximated
up to second order. As a result we obtain an efficient algorithm for sampling
Brownian motion on compact Riemannian manifolds.
