The rate of escape for anisotropic random walks in a tree

Summary Let G be the group generated by L free involutions, whose Cayley graph T is the infinite homogeneous tree with L edges at every node. A general central limit theorem and law of the iterated logarithm is proven for left-invariant random walks Z_n on G or T which applies to the distance of Z_n from a fixed point, as well as to the distribution of the last R letters in Z_n. For nearest neighbor random walks, we also derive a generating function identity that yields formulas for the asymptotic mean and variance of the distance from a fixed point. A generalization for Z_n with a finitely supported step distribution is derived and discussed.Partially supported by grant NSF MCS85-04315