  We present recent results about the asymptotic behavior of ergodic products
of isometries of a metric space X. If we assume that the displacement is
integrable, then either there is a sublinear diffusion or there is, for almost
every trajectory in X, a preferred direction at the boundary. We discuss the
precise statement when X is a proper metric space and compare it with classical
ergodic theorems. Applications are given to ergodic theorems for nonintegrable
functions, random walks on groups and Brownian motion on covering manifolds.
