Random walks on trees and the law of iterated logarithm

Abstract

In this paper we give an alternative proof for the main result of Konsowa and Mitro (J. Theor. Probab. 4 (3) (1991) 535), Konsowa and Mitro found that the simple random walk (SRW) on infinite trees is transient or recurrent. In part of their work, they considered the case of an N -tree in which all the vertices of the same distance n from the root have the same degree which is 3 with probability q n and 2 with probability 1− q n . They proved that the SRW is transient if lim inf nq n>1/ log 2 and recurrent if lim sup nq n<1/ log 2 . We find that the Kolmogorov's law of iterated logarithm is a natural tool to tackle this problem and use it to give an alternative proof.