If are iid random variables and, it is shown that entails either lim or lim or both lim sup and
lim inf. This strengthens somewhat the result of Strassen (1966) asserting that implies lim
sup. Some results on iterated logarithm type behavior for independent but not identically
distributed random variables are also given.

… If EX = 0 and EX2 = u2 < 00, the law of the iterated logarithm (Hartman and Wintner [1])
tells us that … Furthermore, if EX = J1 ¥= 0, the truth of the theorem is evident from the
strong law of large numbers so we may subsequently restrict ourselves to the case EX = O.
We shall proceed by developing a modification of the argument of Stone [3]. Write F for the
distribution function of X. … Then, using the law of the iterated logarithm, …