Last Die Limit

Question

Fix two positive integers $r$ and $n$. A die with sides $1-n$ is formed such that side $i$ appears with probability $p_i$. The die is repeatedly rolled until the sum of all upfaces is at least $r$. For $1 \leq k \leq n$, let $p(k,r)$ be the probability that the last value rolled was $k$. Evaluate $\displaystyle \lim_{r \rightarrow \infty} p(k,r)$. Report the answer when $n = 12, k = 3,$ and $p_i = \frac{6 - |7-i|}{36}$ for each $1 \leq i \leq 12$. This represents the case of two fair dice summed up.