Similar to the central limit theorem, maximum likelihood estimates are asymptotically multivariate normal provided that the model is smooth and log-concave.
Theorem (Asymptotic normality of the MLE): Suppose conditions (A)-(D) are met. Then the maximum likelihood estimator $\bth$ satisfies
\[\sqrt{n}(\bth - \bt^*) \inD \Normal(\zero, \fI(\bt^*)^{-1}).\]Proof: For the MLE, $\u(\bth)=0$. Thus, noting that the MLE is consistent under conditions (A)-(D), the Taylor expansion of the score yields
\[\tfrac{1}{\sqrt{n}} \u(\bts) = \left[ \fI(\bts) + o_p(1) \right] \sqrt{n}(\bth-\bts).\]The term on the left converges in distribution to $N(\zero, \fI(\bts))$, while the term in the square brackets converges to $\fI(\bts)$. Thus, by a minor extension of Slutsky’s theorem, we have
\[\begin{align*} \sqrt{n}(\bth-\bts) &\inD \fI(\bts)^{-1} N(\zero,\fI(\bts)) \\ &=_d N(\zero, \fI(\bts)^{-1}) \end{align*}\]