The rate of escape for anisotropic random walks in a tree

Abstract

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 Zn on G or T which applies to the distance of Zn from a fixed point, as well as to the distribution of the last R letters in Zn. 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 Zn with a finitely supported step distribution is derived and discussed. © 1987 Springer-Verlag.