We discuss about random walks on finitely generated nilpotent groups.
As we see, the behaviour of a random walk at infinity is closely
related to the Gromov-Hausdorff limit of the groups.
We show a large deviation principle for the random walk and see that
it gives a version of a (strong) law of large numbers on groups. This
describes the point to which the random walk converges. We will also
see how the Carnot-Caratheodory metric involves in this limit theorem.