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

Abstract

Edgeworth expansions for non-symmetric 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.