Consider a vertex-reinforced jump process defined on a regular tree, where each vertex has exactly $b$ children, with $b \geq 3$. We prove the strong law of large numbers and the central limit theorem for the distance of the process from the root. Notice that it is still unknown if vertex-reinforced jump process is transient on the binary tree.
"Limit theorems for vertex-reinforced jump processes on regular trees." Electron. J. Probab. 14 1936 - 1962, 2009. https://doi.org/10.1214/EJP.v14-693