= Solution
For $X_1\sim\operatorname{Poisson}(\theta)$, its <probability generating function> is
$$
G(s)=\exp(\theta(s-1)).
$$
Thus $\mathbb E[X_1]=G'(1)=\theta$ and
$$
\operatorname{Var}(X_1)=G''(1)+G'(1)-G'(1)^2=\theta.
$$
The <log-likelihood> is, up to an additive constant,
$$
\ell(\theta)=-n\theta+\left(\sum_{i=1}^nX_i\right)\log\theta,
$$
so the <maximum-likelihood estimator> is $\widehat\theta_{\rm MLE}=\overline X$. The <central limit theorem> gives
$$
\boxed{\sqrt n(\widehat\theta_{\rm MLE}-\theta)
\xrightarrow{d}N(0,\theta).}
$$
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