= Solution
A <maximum-likelihood estimate> $\hat\theta_n$ is a measurable maximizer of $\ell_n(\theta)=\sum_i\log f(X_i,\theta)$ over $\Theta$. Under identifiability, consistency conditions, an interior true parameter, sufficient differentiability and finite positive <Fisher information>, the standard <limits> are
$$
\boxed{\hat\theta_n\xrightarrow{\mathbb P}\theta_0,\qquad
\sqrt n(\hat\theta_n-\theta_0)\xrightarrow{d}\mathcal N(0,I(\theta_0)^{-1}).}
$$
The <variance> in the <normal distribution> is the reciprocal of the per-observation <Fisher information>, not of $nI$. Strong consistency needs additional hypotheses if it is to replace <convergence in probability>; the regular asymptotic result stated here does not require claiming it.
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