Let {X,Xn;n≥1} be a sequence of i.i.d. mean-zero random variables, and let Sn=∑i=1nXi,n≥1. We establish necessary and sufficient conditions for having with probability 1, 0n→∞|Sn|/ $\sqrt{nh(n)}$ <∞, where h is from a suitable subclass of the positive, nondecreasing slowly varying functions. Specializing our result to h(n)=(loglogn)p, where p>1 and to h(n)=(logn)r, r>0, we obtain analogues of the Hartman–Wintner LIL in the infinite variance case. Our proof is based on a general result dealing with LIL behavior of the normalized sums {Sn/cn;n≥1}, where cn is a sufficiently regular normalizing sequence.