I have a process $X_{n+1} = X_n\xi_n$ where $\xi_n\sim\mathcal N(1,1)$ and $\xi_n$ is independent of $X_n$. I need to prove that if $X_0\neq0$ then $ \mathsf P\{|X_n|>1\text{ for some }n\geq0\} = 1. $ From this I construct a random walk: $Y_n = \log|X_n|$ so $ Y_{n+1} = Y_n+\eta_n $ where $\eta_n = \log|\xi_n|$. I guess that from here I should apply the Law of Large Numbers - but I'm stacked with it. Could you help me? For now I should prove that $Y_n$ will eventually be positive a.s. starting from any point.
On the other hand, $X_n$ is a martingale which maybe also useful for deriving the desired result. If it helps, one can take $\xi_n\sim\mathcal N(m,1)$ for some $m\geq1$.