The rate of escape for anisotropic random walks in a tree (Q1085892)

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.