Let Xn for n[ges ]1 be independent random variables with EXn=0 and EX2n=1. Set Sk,n= [sum ]i1<…<ik[les ]nXi1…Xik. Define Tk,c,m= inf{n[ges ]m[ratio ][mid ]k!Sk,n[mid ] >cnk/2}. We study critical values ck,p for k[ges ]2 and p>0, such that ETpk,c,m<∞ for c<ck,p and all m, and ETpk,c,m=∞ for c>ck,p and all sufficiently large m. In particular, c1,1=c2,1=1, c3,1=2 and c4,1=3 under certain moment conditions on X1, when Xn are identically distributed. We also investigate perturbed stopping rules of the form Th,m=inf{n[ges ]m[ratio ] h(S1,n/n1/2) <ζn or >ζn} for continuous functions h and random variables ζn∼a and ζn∼b with a<b. Related stopping rules of the Wiener process are also considered via the Uhlenbeck process.