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 depend 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.

中文翻译：

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覆盖多项式体积增长图的中心随机游走的 Edgeworth 展开

在一些自然假设下获得了具有多项式体积增长组的覆盖图上随机游走的 Edgeworth 扩展。在这个展开式中出现的系数不仅取决于底层图的几何特征，还取决于图在某个幂零李群中的修正谐波嵌入。此外，我们应用 Trotter 逼近定理中的收敛率来建立随机游走的 Berry-Esseen 型边界。