This article examines the rate of escape for a random walk on $G\wr \Z$ and proves laws of the iterated logarithm for both the inner and outer radius of escape. The class of G for which these results hold includes finite, G as well as groups of the form $H\wr \Z$, so the construction can be iterated. Laws of the iterated logarithm are also found for random walk on Baumslag--Solitar groups and a discrete version of the Sol geometry.
"Rate of escape of random walks on wreath products and related groups." Ann. Probab. 31 (4) 1917 - 1934, October 2003. https://doi.org/10.1214/aop/1068646371