Consider the random walk $S_n=\sum_{i=1}^n X_i$ with step size \begin{equation} X_i = \left\{ \begin{array}{} a & \mathrm{with\ probability\ } p \\ b & \mathrm{with\ probability\ } 1-p \\ \end{array} \right. \end{equation} where $a$ and $b$ are numbers which are not integer multiples of each other, or even irrational numbers.
Further assume that $\mathbb{E}X<0$ and $S_0=0$, how can I get the distribution of the maximizer of this random walk? Or to simplify the problem a little bit, what it the probability that the maximizer is at $S_0$?
Edit: Here's a simpler question, is the distribution a scale family? If so, what would be the scale parameter? I think $\mathbb{E}X/\sqrt{Var(X)}$ makes some sense but I was not quite sure, any thoughts on this?