On necessary and sufficient conditions for the law of the iterated logarithm

Let $\{X_n;\,n=1,2,\dots\}$ be a sequence of independent not necessarily identically distributed random variables and $\{a_n;\,n=1,2,\dots\}$ be a non-decreasing sequence of positive numbers such that $a_n\to\infty$. We put $\displaystyle S_n=\sum_{j=1}^n X_j$. Necessary and sufficient conditions are found for the relations $\limsup(S_n/a_n)\le 1$ a.s. and $\limsup(S_n/a_n)=1$ a.s. No assumptions about existence of any moments are made.