Let $(X_i)_{i=1}^n$ be a sequence of i.i.d. random variables with $\Pr(X_i=1)=1-\Pr(X_i=-1)=p$, for some $0\leq p\leq 1$. Let $S_0=0$ and, for $1\le i\le n$, $S_i = X_1+X_2+\cdots+X_i$. Let $\Pi_n$ denote the number of strictly positive terms among $S_1,\ldots,S_n$. In the symmetric case, namely, when $p=1/2$, the law of $\Pi_n$ is known to be (e.g. Feller, Ch. XII.8) $$ \Pr(\Pi_{n}=k) = {2k\choose k}{2n-2k\choose n-k}\frac1{2^{2n}}. $$ Is the law of $\Pi_n$ (even an asymptotic expression in $n$ and $k$) known for the asymmetric case?
Edit: I think that to solve the above general case, it will be suffice to say something about $$ \Pr(\Pi_{n}=n) $$ which is perhaps simpler, but my attempts failed.