Limit Theorems Without Moment Hypotheses for Sums of Independent Random Variables

Abstract

Let $\{S_n\}$ be the partial sums of a sequence of independent random variables and let $\{a_n\}$ be a nondecreasing, divergent real sequence. Necessary and sufficient conditions for $\lim \sup_{n\rightarrow\infty}S_n/a_n < \infty$ a.s. are given under mild conditions on $\{S_n\}$; these conditions do not involve the existence of any moments. These results are employed to widen the scope of the law of the iterated logarithm.