Hi !

Okay, this one has been bugging (and p***ing) me...

We have $\displaystyle (Y_i)_i$ iid rv's where $\displaystyle P(Y_1=1)=p$ and $\displaystyle P(Y_1=-1)=1-p$

Let $\displaystyle a\in\mathbb{N}^*$ and $\displaystyle S_n=a+\sum_{i=1}^n Y_i$

So for $\displaystyle p=1/2$, $\displaystyle S_n$ is a martingale (with the natural filtration)

Let $\displaystyle S_{n\wedge T}=S^T$ be the stopped martingale, where $\displaystyle T=\inf\{n>0 ~:~ S_n=0 \text{ or } S_n=a+b\}$ (b is another positive integer)

Then I have to prove, using the convergence of $\displaystyle S^T$, that $\displaystyle P(T<\infty)=1$.

And I don't know how to do it (I've been scrabbling on it, without finding a formal proof...).

Thanks for any hint...