A note on recurrent random walks

For any recurrent random walk (Sn)n⩾1 on R, there are increasing sequences (gn)n⩾1 converging to infinity for which (gnSn)n⩾1 has at least one finite accumulation point. For one class of random walks, we give a criterion on (gn)n⩾1 and the distribution of S1 determining the set of accumulation points for (gnSn)n⩾1. This extends, with a simpler proof, a result of Chung and Erdös. Finally, for recurrent, symmetric random walks, we give a criterion characterizing the increasing sequences (gn)n⩾1 of positive numbers for which lim̲gn|Sn|=0.