Asymptotic properties of MLEs Are there any relationship between the asymptotic properties of MLEs (assuming that the regularity conditions hold)? 
I mean, once I know that the MLE for $\tau(\theta)$ is asymptotically efficient and asymptotically normally distributed, how can I explain why $\tau(\hat{\theta})$ converges in probability to $\tau(\theta)$? Which results should I use to prove this convergence?
 A: There is a separate proof concerning the consistency of the mle but if you take the asymptotic normality for granted, that is if you want to start from:
$$\sqrt{n} \left(\hat{\theta} - \theta \right) \xrightarrow{D} N \left(0, I(\theta)^{-1} \right)$$
(and similarly for a function of $\theta$ under the Delta method), you can just use the fact that the variance of $\hat{\theta}$ goes to zero under the regularity conditions on the Fisher Information, namely that it exists and that it is bounded. Since mean-squared error consistency implies convergence in probability, you are done.
The original proof of consistency relies on Jensen's inequality to show that the true parameter asymptotically maximizes the likelihood. Then since the mle is itself a maximizer, under some regularity conditions it can be shown that it will be arbitrarily close to the true parameter as the sample grows larger.
A: To generalize a bit from MLEs to general asymptotic normality results, if you know that
$$
\sqrt{n}(\hat\theta_n-\theta)\to_d N(0,Avar(\hat\theta_n))
$$
it generally follows that
$$
\sqrt{n}(\hat\theta_n-\theta)=\mathcal{O}_p(1)
$$
and thus 
$$
\hat\theta_n-\theta=\mathcal{O}_p(n^{-1/2})=o_p(1)
$$
so that 
$$
\hat\theta_n\to_p\theta
$$
So if you have proven asymptotic normality, you have also proven consistency. (In the case of some proofs like exactly the MLE case, you must however take care to avoid circular reasoning, as proofs of asymptotic normality of the MLE assume consistency as a preliminary result.)
