When you characterize this as a probability question rather than a
general series question, if you avoid certain pitfalls you can
use the Law of Large Numbers to characterize the sum.
That is, having defined a probability distribution for $a_k$,
this implies a probability distribution of $\log(a_k)$;
then from the expectation and variance of $\log(a_k)$
you can estimate the expectation and variance of the sum for large $n$.
There is a discussion of the case where $a_k$ is (approximately)
normally distributed on stats.SE
(see this question)
and a discussion of the case where $a_k$ has a binomial distribution
on mathoverflow (see this question).
One technique that is recommended is to use the Taylor series of
$\log(x)$ centered at $\mathbb E(a_k)$.
You cannot actually let $a_k$ have a normal distribution, because
any normal distribution has negative values.
In fact, you really want to prevent $a_k$ from getting close to zero,
because the resulting large negative logarithms will cause problems.
(This is one of the above-mentioned "pitfalls" to avoid.)
A binomial distribution with only positive integer values
would produce a much better-behaved distribution of the logarithm.