Edgeworth expansions for centered random walks on covering graphs of polynomial volume growth

TL;DR

This paper derives Edgeworth expansions for centered random walks on covering graphs with polynomial volume growth, linking geometric and harmonic features, and establishes Berry-Esseen bounds for convergence rates.

Contribution

It introduces Edgeworth expansions for such random walks considering geometric and harmonic embedding factors, and applies Trotter's theorem for convergence rate analysis.

Findings

Edgeworth expansions depend on geometric and harmonic features.

Established Berry-Esseen bounds for the convergence of random walks.

Applied Trotter's approximation theorem to rate of convergence.

Abstract

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.