A note on recurrent random walks

Abstract

For any recurrent random walk ( S n ) n ⩾ 1 on R , there are increasing sequences ( g n ) n ⩾ 1 converging to infinity for which ( g n S n ) n ⩾ 1 has at least one finite accumulation point. For one class of random walks, we give a criterion on ( g n ) n ⩾ 1 and the distribution of S 1 determining the set of accumulation points for ( g n S n ) 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 ( g n ) n ⩾ 1 of positive numbers for which lim ̲ g n | S n | = 0 .