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. 



General ballot theorem
You might want to check out the following nice [survey](http://cgm.cs.mcgill.ca/~reedbook/papers/BallotTheorems.pdf) 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. 


