Limited Hitter IV

A $n-$sided die with the values $1-n$ is rolled infinitely. The side with value $i$ appears with probability $p_i > 0$. The cumulative sum of all observed rolls is recorded after each roll. For each integer $k \geq 1$, let $f(k)$ be the probability that the cumulative sum is ever exactly $k$. For a fixed $k$, compute $\displaystyle \lim_{k \rightarrow \infty} f(k)$. Report the answer when $n = 8, p_i = 1/12$ for $i = 1,2,3,4$, and $p_i = 1/6$ for $i = 5,6,7,8$.