Law of large numbers with weights

Law of large numbers with weights

Let $X_1 , X_2 , ...$ be an iid sequence with mean 0 and finite variance.
Let $(a_n)$ be a sequence of non-random real numbers. Under what
conditions on $(a_n)$ do the weighted means $\frac{1}{n} \sum_{j=1}^n a_j
X_j$ converge almost surely to 0?
The law of large numbers tells us that this will be the case if $a_j = 1$
for each $j$. By scaling the same is true if each $a_j$ is equal to the
same constant $c$. Furthermore, if $c \leq a_j \leq C$ for each $j$, then
we have
$$ \frac{c}{n} \sum_{j=1}^n a_j X_j \leq \frac{1}{n} \sum_{j=1}^n a_j X_j
\leq \frac{C}{n} \sum_{j=1}^n a_j X_j $$
and the left and right sides tend to 0 a.s. So, the statement is true if
the $a_j$'s all belong to some finite interval. However, it strikes me as
unlikely to be true if the $a_j$'s grow too rapidly. So, I am looking for
some minimal condition under which the convergence does hold. Does anyone
know of one?