Let Aff(𝕋) be the group of isometries of a homogeneous tree 𝕋 fixing an end of its boundary. Given a probability measure on Aff(𝕋) we consider an associated random process on the tree. It is known that under suitable hypothesis this random process converges to the boundary of the tree defining a harmonic measure there. In this paper we study the asymptotic behaviour of this measure.

The convergence rate of the expectation of the logarithm of the first return time $R_n$, after being properly normalized, is investigated for ergodic Markov chains. I. Kontoyiannis showed that for any β > 0 we have $log\left[R_n\left(x\right)P_n\left(x\right)\right]=o\left(n^{\beta }\right)$ a.s. for aperiodic cases and A. J. Wyner proved that for any ε >0 we have $-(1+\epsilon )logn\le log\left[R_n\left(x\right)P_n\left(x\right)\right]\le loglogn$ eventually, a.s., where $P_n\left(x\right)$ is the probability of the initial n-block in x. In this paper we prove that $E[log{R}_{(L,S)}-(L-1)h]$ converges to a constant depending only on the process where $R_{(L,S)}$ is the modified first return time with...