= Finite-exposure Poisson inconsistency
{title2=$\widehat\theta_n\to S_\infty/a_\infty$}
Let independent counts satisfy $Y_i\sim\operatorname{Poi}(\theta w_i)$, with positive known exposures $w_i$, $\theta>0$, and $a_n=\sum_{i=1}^nw_i\uparrow a_\infty<\infty$. Their <likelihood> is proportional to $\theta^{S_n}e^{-a_n\theta}$, where $S_n=\sum_{i=1}^nY_i$. If $S_n>0$, the positive <maximum-likelihood estimator> is $S_n/a_n$; when $S_n=0$, no maximum is attained on the open parameter space, while the closed-space maximizer is zero. This extended estimate converges almost surely to $S_\infty/a_\infty$, where $S_\infty\sim\operatorname{Poi}(\theta a_\infty)$. Finiteness follows from $\mathbb E S_\infty=\theta a_\infty$, and the limiting <Poisson distribution> follows from the finite-sum laws. In particular, the probability of a zero estimate tends to $e^{-\theta a_\infty}>0$, proving failure of <statistical consistency>. The total <Fisher information> tends to $a_\infty/\theta$: infinitely many observations need not provide infinite information.
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