Let $(X_k)_{k\ge 1}$ be i.i.d random variables, define $S_n=X_1+X_2+\cdots+X_n, \:S_0=0$ and assume that $(S_n)$ can only jump down by $-1$. For a certain $\lambda$ consider the martingale $M_n=e^{-\lambda S_n}$. Let $T=\inf(n\ge0,S_n=-1)$ be the hitting time of $-1$ by $S_n$.
It is a multi part question and I am really stuck on the cases when the hitting time is infinity since Doob's theorem doesn't apply and I don't know how to approach them.
*If $T= \infty$, what is the limit of the stopped process $(S^T)$ as $n\to +\infty$?
*If $T= \infty$, what is the limit of the stopped Martingale $(M^T)$ as $n\to +\infty$
Amy help would be appreciated,
thanks
We exclude the trivial cases by assuming that $\mathbb{P}(X_1<0)>0$ and that $\lambda>0$. As an exercise, you can easily prove that $$ \mathbb{P}(\{\liminf S_n<\infty\}\cap\{T=\infty\})=0. $$ Once done, you immediately deduce that, if $T=\infty$, then $S^T_n\rightarrow \infty$ and $M^T_n\rightarrow 0$.