Consider the Markov chain $p(n_{i+1}\mid n_i)$ where $n_0 = 1$, and $n_{i+1}$ is poisson distributed with mean $\lambda n_i$, that is $$ p(n\mid k) = \frac{(k \lambda)^n \mathrm{e}^{-k \lambda}}{n!}. $$ Is there a name for this process, or a literature surrounding it?
In particular I am interested in the properties of the distribution $P(N)$ of the quantity $N = \sum_i n_i$ in the case where $\lambda$ is sufficiently small that $P(N) \to 0$ as $N \to \infty$.
As I expect an exponentially rare subset of outcomes to contribute and exponentially large amount to the mean, I presume that $P(N) \sim N^{-\alpha}$ at large $N$. I would be interested to verify this, and to calculate $\alpha$ in terms of $\lambda$.