= Bounded information in a subcritical Poisson branching process
{title2=$i_n(\theta)=\frac{1-\theta^{n+1}}{\theta(1-\theta)},\quad0<\theta<1$}
Start with $Y_0\sim\operatorname{Poisson}(\theta)$ and conditionally let $Y_i\sim\operatorname{Poisson}(\theta Y_{i-1})$. 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 $S_n=\sum_{i=0}^nY_i$ 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
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