Let (X, $X_{n}$, n$geq$1) be a sequence of i.i.d. random variables and { $a_{ni}$ , 1$leq$i$leq$n, n$geq$1} be an array of constants. Let ø($chi$) be a positive increasing function on (0, $infty$) satisfying ø($chi$) ↑ $infty$ and ø(C$chi$) = O(ø($chi$)) for any C > 0. When EX = 0 and E[ø(｜X｜)]〈$infty$, some conditions on ø and { $a_{ni}$ } are given under which (equation omitted).).