Suppose we have the following random walk. Let $X_0=\lambda \in \mathbb{N}$ and then
$ \begin{cases} X_t=\operatorname{Pois}(1) & \text{if } X_{t-1}=0 \\ X_t=\operatorname{Pois}(X_{t-1}) & \text{if } 1 \leq X_{t-1} \leq \lambda \\ X_t=\operatorname{Pois}(\lambda) & \text{if } X_{t-1}>\lambda \end{cases} $
From running simulations, it appears that for $t,λ$ large, $X_t$ is very rarely $>λ$. I'd like to prove something precise along these lines. Maybe that, as $t,\lambda \to \infty$, $\mathbb{P}(X_t>\lambda) \to 0$. (but I'm not wedded to this exact question, I'm more interested in learning how one would go about analyzing this, rather than answering a specific question)
I also posted this to stats.stackexchange, but I think it may be a better fit here.