 Bounded information in a subcritical Poisson 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