We prove a central limit theorem for the transition operator of the symmetric random walk on a covering graph with a covering transformation group of polynomial growth. As the limit, the continuous semigroup of the sub-Laplacian on a nilpotent Lie group is obtained.
Satoshi ISHIWATA. "A central limit theorem on a covering graph with a transformation group of polynomial growth." J. Math. Soc. Japan 55 (3) 837 - 853, July, 2003. https://doi.org/10.2969/jmsj/1191419005