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Let {S d (n)} n≥0 be the simple random walk inZ d , and Π(d)(a,b)={S d (n)∈Z d :a≤n≤b}. Supposef(n) is an integer-valued function and increases to infinity asn tends to infinity, andE n (d) ={Π(d)(0,n)∩Π(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.
Acta Mathematicae Applicatae Sinica – Springer Journals
Published: Jul 13, 2005
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