Let $X_{s}$ be a Bernoulli r.v such that it returns (1-p) with probability p and returns (-p) with probability (1-p). All $X_s$ are mutually independent. Let $S_n = X_1 + X_2 + \cdots X_n$ and T is the stopping time when $S_n<a$ or $S_n>b$ for some $a<0$ or $b>0$.
- What is the distribution of the stopping time T? I guess expected stopping time might be infinity as a random walk does, but I am not sure.
- Now consider the unbalanced case, so that $X_s$ returns (1-p) with probability q and returns (-p) with probability (1-q). WLOG $q>p$. Now can we calculate the distribution of the stopping time T?
I have read some documents about discrete random walks, and it uses characteristic function to calcualte expected stopping time, but I don't think I can apply that method to this problem.
I am now researching about CUSUM(CUmulative SUM), SPRT(Sequential Probability Ratio Test) things and it is necessary for my further research - I need more information than upper bound of stopping time...... Please help......