Let $\lambda_0=10^5$ or any other large integer. Define the recursive "process": $\lambda_t=\text{sample from a Poisson distribution with mean }\lambda_{t-1}$. Is this process recurrent? I mean, after a long time, does the probability of reaching "the other side" of the initial mean have a non-zero lower bound? Or is the probability of reaching 0 or $\infty$ positive?
Thanks!
I did some initial experiments (t=20k, repeat for 10 times, all decreasing) finding that the mean has trend to decrease. According to wiki, Poisson with large mean can be approximated by normal($\lambda,\lambda$). By normal approximation, I found that the occurrence of $\lambda_{20k}<10k$ is slightly more than that of $\lambda_{20k}>10k$ (12 vs 8).