An intersection property of the simple random walks inZ d

Let {S_d(n)}_n0 be the simple random walk inZ^d, and ^(d)(a,b)={S_d(n)Z^d:anb}. Supposef(n) is an integer-valued function and increases to infinity asn tends to infinity, andE_n^(d)={^(d)(0,n)^(d)(n+f(n),)}. In this paper, a necessary and sufficient condition to ensureP(E_n^d),i.o.)=0, or 1 is derived ford=3, 4. This problem was first studied by P. Erdös and S.J. Taylor.This work is partly supported by the National Natural Sciences Foundation of China.