An exponential limit distribution is obtained for stopping times associated with partial sums of independent, identically distributed random variables whose distribution function is slowly varying at infinity. It is also demonstrated that a generalized law of the iterated logarithm cannot obtain in such a case.
"Rapidly Growing Random Walks and an Associated Stopping Time." Ann. Probab. 7 (6) 1078 - 1081, December, 1979. https://doi.org/10.1214/aop/1176994903