A Universal One-Sided Law of the Iterated Logarithm

Abstract

We prove that the $\lim \inf$ of suitably normalized sums of i.i.d. nonnegative and nondegenerate random variables can with probability 1 only be a constant between $-2^{1/2}$ and 0. Moreover, we show that each value within this range is attainable by an appropriate choice of the underlying common distribution function.