A general integral test is established which refines the Jain-Pruitt C
hung LIL for iid random variables. As a corollary we obtain that Chung
's integral test for Brownian motion is valid for partial sums of iid
random variables satisfying EX(2)1{Absolute value of X greater-than-or
-equal-to t} = O((LLt)-1) as t --> infinity.