Asymptotic behaviour of $P(S_n=k),$ where $S_n$ is a sum of $n$ i.i.d. discrete random variables and $k$ is fixed. Let $\{X_i\}_{i=1}^\infty$ be a sequence of i.i.d. discrete random variables taking non-negative integer values and let $S_n=\sum_{i=1}^nX_i$. Are there any theorems/results establishing exact asymptotic behaviour (say, in terms of mean and/or variance of $X_i$'s) of $$P(S_n=k)$$
as $n\to\infty,$ where $k\in\mathbb{N}_0$ is fixed?
 A: I'll assume $k > 0$.
There are only finitely many partitions of $k$, i.e. ways to obtain $k$ as a sum of positive integers.  For any such partition ${\bf a} = (a_1, \ldots, a_m$) where $a_1 + \ldots + a_m = k$, the probability
that $(X_1, \ldots, X_n)$ is a permutation of $(a_1, \ldots, a_m, 0, \ldots, 0)$ is
$$ P_n({\bf a}) = \frac{n!}{(n-m)! \prod_j c_j!} \mathbb P(X=0)^{n-m} \prod_{i=1}^m \mathbb P(X = a_i) $$ where $c_j$ are the multiplicities of the distinct values of $\bf a$.  We then have $\mathbb P(S_n=k) = \sum_{\bf a} P_n({\bf a})$.
As $n \to \infty$ we have $$P_n({\bf a}) \sim \text {const}\; n^m \mathbb P(X=0)^{n-m}$$
with a nonzero constant iff $\mathbb P(X=0)$ and $\mathbb P(X=a_i)$ are all nonzero.  If $\mathbb P(X=0)$ and $\mathbb P(X=1)$ are nonzero, 
the dominant term is for ${\bf a} = (1,\ldots,1)$ which has $m=k$ and $c_1 = k$.  Thus
$$\mathbb P(S_n = k) \sim \frac{n^k \mathbb P(X=0)^{n-k} \mathbb P(X=1)^k}{k!}$$
If $\mathbb P(X=1) = 0$, it's more complicated: you have to find ways to partition $k$ into as many parts as possible where each part has nonzero probability.
