We show that the group of bounded automatic automorphisms of a rooted tree is amenable, which implies amenability of numerous classes of groups generated by finite automata. The proof is based on reducing the problem to showing amenability just of a certain explicit family of groups (mother groups) which is done by analyzing the asymptotic properties of random walks on these groups.
"On amenability of automata groups." Duke Math. J. 154 (3) 575 - 598, 15 September 2010. https://doi.org/10.1215/00127094-2010-046