On the average of a random walk

Let {S_n, n ? N)} be a simple random walk and denote by A_n its time average: A_n = (S₁+ …+S_n)/n. We give an integral test for the lower bound on A_n, thus giving an affirmative answer to a conjecture of P. Erdös (private communication) that A_n will return to a fixed region around the origin infinitely often with probability 1 in 1 dimension whereas in 2 or more dimensions it will return only finitely many times.