Simple condition for asymptotic normality of MLE Are these conditions sufficient for asymptotic normality of MLE? If it is, pls. let me know the references.
1-First and second derivatives of $\ell(\theta,\eta)$ are defined. 
2- The Fisher information matrix be non-singular and continuous with respect to the parameters $\theta$ and $\eta$.
 A: Asymptotic normality is usually proven for a local maximum of the likelihood function. I paste below the conditions as stated in T. Amemiya (1985) Advanced Econometrics, ch. 4, for extremum or $M$-estimators in general. $T$ is sample size, $\theta$ is the unknown parameter vector, and  $\theta_0$ is the true values, $\mathbf y$ is the data sample, $Q_T(\mathbf y, \theta)$ is the objective function, $\hat \theta_T$ is the ML estimator. $\Theta_T$ is the set containing the roots of the equation $\frac {\partial Q_T(\mathbf y, \theta)}{\partial \theta} =0$ corresponding to local maxima.

Variations do exist. For example, if $Q_T(\mathbf y, \theta)$ is concave over the parameter space for any $\mathbf y$, then uniform convergence (4.1.2.-C) can be replaced by point-wise convergence. Also, there are alternative ways to express some of the conditions. For example condition 4.1.3-B may be replaced by its own sufficient condition, the local dominance condition : for some neighborhood of $\theta_0$, $N(\theta_0)$,
$$E\left[\text {sup}_{\theta \in N(\theta_0)}  ||\partial^2Q_T/\partial \theta\partial\theta'||\right] <\infty$$.
ADDENDUM
Responding to @whuber's comment (my meta-response can be found in the comments), consider an i.i.d sample $\{x_i,\, i=1,...,n\}$ from a uniform $U(0,\theta)$. Then the ML estimator for $\theta$ is
$$\hat \theta_{ML} = \max_i x_i = X_{(n)}$$
The asymptotic distribution of the maximum of an i.i.d sample is known not to be normal. This is an example that asymptotic normality requires more conditions than those stated in the question.
