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

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 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.

Appendix A. Proof of Proposition 4.2

We here give a proof of Proposition 4.2 in the case $N=2$.

Proof of Proposition 4.2

Throughout the proof, $\langle \cdot , \cdot \rangle _{\ell ^2(X_0)}$ and $\Vert \cdot \Vert _{\ell ^2(X_0)}$ are abbreviated as $\langle \cdot , \cdot \rangle$ and $\Vert \cdot \Vert$, respectively. By virtue of the decomposition (4.1), we have

$$\begin{aligned} \begin{aligned}&\frac{1}{n^2}\sum _{k=0}^{n-1}\sum _{\ell =0}^k\mathcal {L}^\ell f(x)\mathcal {L}^{k+1}g(x)\\&\quad =\frac{1}{2}\Big (1+\frac{1}{n}\Big )\langle f, m\rangle \langle g, m\rangle + \frac{1}{n^2}\langle f, m\rangle \sum _{k=1}^{n}k\sum _{j=1}^{K_0-1}\langle g, \psi _j\rangle \alpha _j^{k}\phi _j(x)\\&\qquad +\,\frac{1}{n^2}\langle f, m\rangle \sum _{k=0}^{n-1} \sum _{\ell =0}^k \mathcal {L}^{\ell } g_{\ell _{K_0}^2(X_0)}(x)+\frac{1}{n^2}\langle g, m\rangle \sum _{k=0}^{n-1}\sum _{\ell =0}^k \sum _{j=1}^{K_0-1}\langle f, \psi _j\rangle \alpha _j^{\ell }\phi _j(x)\\&\qquad +\,\frac{1}{n^2}\sum _{k=0}^{n-1}\sum _{\ell =0}^k \Big (\sum _{j=1}^{K_0-1}\langle f, \psi _j\rangle \alpha _j^{\ell }\phi _j(x)\Big ) \Big (\sum _{j=1}^{K_0-1}\langle g, \psi _j\rangle \alpha _j^{k+1}\phi _j(x)\Big )\\&\qquad +\,\frac{1}{n^2}\sum _{k=0}^{n-1}\mathcal {L}^{k+1}g_{\ell ^2_{K_0}(X_0)}(x) \sum _{\ell =0}^k \Big (\sum _{j=1}^{K_0-1}\langle f, \psi _j\rangle \alpha _j^{\ell }\phi _j(x)\Big )\\&\qquad +\,\frac{1}{n^2}\langle g, m\rangle \sum _{k=1}^{n}k \mathcal {L}^{k}f_{\ell ^2_{K_0}(X_0)}(x)\\&\qquad +\,\frac{1}{n^2}\sum _{k=0}^{n-1} \Big (\sum _{j=1}^{K_0-1}\langle g, \psi _j\rangle \alpha _j^{k+1}\phi _j(x)\Big ) \sum _{\ell =0}^k \mathcal {L}^{\ell }f_{\ell ^2_{K_0}(X_0)}(x)\\&\qquad +\,\frac{1}{n^2}\sum _{k=0}^{n-1}\sum _{\ell =0}^k \mathcal {L}^{\ell }f_{\ell ^2_{K_0}(X_0)}(x) \mathcal {L}^{k+1}g_{\ell ^2_{K_0}(X_0)}(x)\\&\quad =:\frac{1}{2}\Big (1+\frac{1}{n}\Big )\langle f, m\rangle \langle g, m\rangle +I_1+I_2+I_3+I_4+I_5+I_6+I_7+I_8 \end{aligned} \end{aligned}$$

for $n \in \mathbb {N}$ and $x \in V_0$. In particular, the terms $I_1, I_3, I_4$ and $I_5$ are calculated as follows:

$$\begin{aligned} \begin{aligned} I_1&=-\frac{1}{n}\langle f, m \rangle \sum _{j=1}^{K_0-1}\frac{\alpha _j^{n+1}\langle g, \psi _j\rangle }{1-\alpha _j}\phi _j(x)+\frac{1}{n^2}\langle f, m \rangle \sum _{j=1}^{K_0-1}\frac{\alpha _j(1-\alpha _j^n)\langle g, \psi _j\rangle }{(1-\alpha _j)^2}\phi _j(x),\\ I_3&=\frac{1}{n}\langle g, m \rangle \sum _{j=1}^{K_0-1}\frac{\langle f, \psi _j\rangle }{1-\alpha _j}\phi _j(x) -\frac{1}{n^2}\langle g, m \rangle \sum _{j=1}^{K_0-1}\frac{\alpha _j(1-\alpha _j^n)\langle f, \psi _j\rangle }{(1-\alpha _j)^2}\phi _j(x),\\ I_4&=\frac{1}{n^2}\sum _{i, j=1}^{K_0-1}\frac{\langle f, \psi _i\rangle \langle g, \psi _j\rangle }{1-\alpha _i}\Big ( \frac{\alpha _j(1-\alpha _j^n)}{1-\alpha _j} - \frac{\alpha _i\alpha _j(1-\alpha _i^n\alpha _j^n)}{1-\alpha _i\alpha _j}\Big )\phi _i(x)\phi _j(x),\\ I_5&=\frac{1}{n^2}\sum _{k=0}^{n-1}\mathcal {L}^{k+1}g_{\ell ^2_{K_0}(X_0)}(x) \sum _{j=1}^{K_0-1}\frac{\langle f, \psi _j\rangle (1-\alpha _j^{k+1})}{1-\alpha _j}\phi _j(x). \end{aligned} \end{aligned}$$

We note that the Perron–Frobenius theorem implies that there exists some $\lambda \in (0, 1]$ such that $\big \Vert \mathcal {L}|_{\ell ^2_{K_0}(X_0)}\big \Vert \le \lambda$. Hence, we see that

$$\begin{aligned} \Big \Vert \sum _{k=0}^{n-1}\mathcal {L}^{k}f\Big \Vert&\le \sum _{k=0}^{n-1} \lambda ^k \Vert f\Vert \le \frac{\Vert f\Vert }{1-\lambda }=O(1), \nonumber \\ \Big \Vert \sum _{k=1}^{n}k\mathcal {L}^{k}f\Big \Vert&\le n\sum _{k=1}^{n} \lambda ^k \Vert f\Vert =O(n), \quad \text {and}\nonumber \\ \Big \Vert \sum _{k=0}^{n-1}\sum _{\ell =0}^k \mathcal {L}^{\ell }f\Big \Vert&\le \sum _{k=0}^{n-1}\sum _{\ell =0}^k \lambda ^k \Vert f\Vert =O(n) \end{aligned}$$

(A.1)

for $f \in \ell _{K_0}^2(X_0)$. We now set

$$\begin{aligned} A[f, g]_n^{(1)}(x)&= \frac{1}{2}\langle f, m \rangle \langle g, m \rangle -\langle f, m \rangle \sum _{j=1}^{K_0-1}\frac{\alpha _j^{n+1}\langle g, \psi _j\rangle }{1-\alpha _j}\phi _j(x) \nonumber \\&\quad +\,\frac{1}{n}\langle f, m\rangle \sum _{k=0}^{n-1} \sum _{\ell =0}^k \mathcal {L}^{\ell } g_{\ell _{K_0}^2(X_0)}(x)\nonumber \\&\quad +\,\langle g, m \rangle \sum _{j=1}^{K_0-1}\frac{\langle f, \psi _j\rangle }{1-\alpha _j}\phi _j(x) +\frac{1}{n}\langle g, m\rangle \sum _{k=1}^{n}k \mathcal {L}^{k}f_{\ell ^2_{K_0}(X_0)}(x), \end{aligned}$$

(A.2)

and

$$\begin{aligned} A[f, g]_n^{(2)}(x)&= \langle f, m \rangle \sum _{j=1}^{K_0-1}\frac{\alpha _j(1-\alpha _j^n)\langle g, \psi _j\rangle }{(1-\alpha _j)^2}\phi _j(x)\nonumber \\&\quad -\langle g, m \rangle \sum _{j=1}^{K_0-1}\frac{\alpha _j(1-\alpha _j^n)\langle f, \psi _j\rangle }{(1-\alpha _j)^2}\phi _j(x) \nonumber \\&\quad +\,\sum _{i, j=1}^{K_0-1}\frac{\langle f, \psi _i\rangle \langle g, \psi _j\rangle }{1-\alpha _i}\Big ( \frac{\alpha _j(1-\alpha _j^n)}{1-\alpha _j} - \frac{\alpha _i\alpha _j(1-\alpha _i^n\alpha _j^n)}{1-\alpha _i\alpha _j}\Big )\phi _i(x)\phi _j(x)\nonumber \\&\quad +\,\sum _{k=0}^{n-1}\mathcal {L}^{k+1}g_{\ell ^2_{K_0}(X_0)}(x) \sum _{j=1}^{K_0-1}\frac{\langle f, \psi _j\rangle (1-\alpha _j^{k+1})}{1-\alpha _j}\phi _j(x) \nonumber \\&\quad +\,\sum _{k=0}^{n-1} \Big (\sum _{j=1}^{K_0-1}\langle g, \psi _j\rangle \alpha _j^{k+1}\phi _j(x)\Big ) \sum _{\ell =0}^k \mathcal {L}^{\ell }f_{\ell ^2_{K_0}(X_0)}(x)\nonumber \\&\quad +\,\sum _{k=0}^{n-1}\sum _{\ell =0}^k \mathcal {L}^{\ell }f_{\ell ^2_{K_0}(X_0)}(x) \mathcal {L}^{k+1}g_{\ell ^2_{K_0}(X_0)}(x). \end{aligned}$$

(A.3)

By noting (A.1), $|\alpha _j|\le 1, \, j=1, 2, \dots , K_0-1$, and an inequality $\Vert fg\Vert ^2 \le |V_0|\Vert f\Vert ^2\Vert g\Vert ^2$ for $f, g \in \ell ^2(X_0)$, we conclude that $\big \Vert A[f, g]_n^{(1)}\big \Vert =O(1)$ and $\big \Vert A[f, g]_n^{(2)}\big \Vert =O(1)$ as $n \rightarrow \infty$. This completes the proof of Proposition 4.2. $\square$