Edgeworth Expansions for Centered Random Walks on Covering Graphs of Polynomial Volume Growth

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 not only geometric features the underlying but also modified harmonic embedding graph into certain nilpotent Lie group. Moreover, we apply rate convergence Trotter's approximation theorem to establish Berry-Esseen type bound walks.