I am looking for a version of the Ballot Theorem for general step distributions. Specifically, let $X_1,X_2,\ldots$ be i.i.d. real random variables with some distribution. Let $S_n = S_1 + \cdots + S_n$. Let $$ p_n = \mathbb{P}[S_1 > 0,\ldots,S_n > 0].$$
For $X_n$ supported on $\{+1,-1\}$ this can be calculated exactly using the Ballot theorem. What can be said more generally? Say even for finitely supported $X_n$? Asymptotic behavior will suffice.
You might want to check out the following nice survey of Ballot theorems by Addario-Berry and Reed. For example, Theorem 3 (due to Takács) deals with the case that each $X_i$ is integer valued with mean $\mu$ and maximum value $1$. As you only care about asymptotic behaviour, Theorem 3 gives a very satisfying answer in this case. That is, $p_n$ converges to $\mu$, if $\mu$ is positive, and otherwise $p_n$ converges to $0$. There are also versions for real-valued random variables. See for example Theorem 7 due to Kallenberg.
