Let $\{\epsilon_t\}_{t\ge0}$ be an iid sequence of random variables and let $\lambda>1$. I am interested in the following process: Let $X_0 = 0$ and $$ X_{t+1} = \lambda(X_t+\epsilon_t). $$ This process goes to $\pm\infty$ with probability one if $\mathbb{E}\log|\epsilon_t|<\infty$. I am specifically interested in the probability that the process ever crosses the origin again. That is, the probability that there exists a $t$ such that $X_t<0$ and $X_{t+1}>0$, or the other way around. I am new at these sort of problems. Can someone point me to some literature that deals with similar questions?
Thank you in advance!