Poisson branching process ($f(s)=e^{\mu (s-1)}$)

A Galton-Watson process with a Poisson distribution of offspring of mean $\mu$ has this probability generating function. For $\mu =1+\epsilon >1$ its branching survival probability $r$ satisfies $-\log (1-r)=(1+\epsilon )r$, and $2\epsilon /(1+2\epsilon )\le r\le 2\epsilon$. Indeed $\epsilon ={\sum }_{j\ge 1}{r}^j/(j+1)$ lies between $r/2$ and $r/[2(1-r)]$. These exact inequalities belong to the Poisson branching process, not to every finite binomial branching process.

Start with $Y_0\sim \mathrm{Poisson}(\theta )$ and conditionally let $Y_i\sim \mathrm{Poisson}(\theta Y_{i-1})$. The mean generation size is ${\theta }^{i+1}$. The log-likelihood is $S_n\log \theta -\theta T_n$ plus a parameter-free term, where $S_n={\sum }_{i=0}^n{Y}_i$ and $T_n=1+{\sum }_{i=0}^{n-1}{Y}_i$. Expected negative curvature gives the displayed Fisher information, bounded as the horizon grows. Extinction makes the observations eventually all zero, so one longer observed family does not supply indefinitely increasing information.

// Target: convergence-of-random-variables.bigb