I have $X = \sum_{i = 0}^n x_i$, where each $x_i$ is a die roll that takes on any of the values $[1, 2, 3, 4, 5, 6]$ with equal probability.
I think that, according to the C.L.T, at $n \rightarrow \infty$ the variable $X$ will become normally distributed.
What I'm wondering: is there any proof of the Central Limit Theorem for this special case -- perhaps more accessible than the general proof?
What I've tried: If we allow dice with a large number of sides to simplify things, I think the number of ways to roll $k$ die and sum to $m$ is $\binom{m - 1}{k - 1}$ (stars-and-bars). The total number of ways to roll the die will be $k^n$, so:
$$ P(X = m) = \frac{\binom{m - 1}{k - 1}}{k^n} $$
But this becomes zero as $n \rightarrow \infty$.