  Edgeworth expansions for random walks on covering graphs with groups of
polynomial volume growths are obtained under a few natural assumptions. The
coefficients appearing in this expansion depends on not only geometric features
of the underlying graphs but also the modified harmonic embedding of the graph
into a certain nilpotent Lie group. Moreover, we apply the rate of convergence
in Trotter's approximation theorem to establish the Berry-Esseen type bound for
the random walks.
