Maximum Likelihood estimates follow a normal distribution

I was quite surprised when I learnt that a maximum likelihood estimate follows asymptotically a normal distribution with the mean being the estimated value and the variance being the inverse of the Fisher Information multiplied by the number of observations.

Asymptotically means when you have a big number of samples to calculate your ML estimate.

This is quite a remarkable fact. Some information:

http://math.arizona.edu/~jwatkins/o-mle.pdf (page 10)

http://people.missouristate.edu/songfengzheng/Teaching/MTH541/Lecture%20notes/Fisher_info.pdf

http://stats.stackexchange.com/questions/88481/what-is-meant-by-the-standard-error-of-a-maximum-likelihood-estimate

http://stats.stackexchange.com/questions/113860/how-to-compute-or-numerically-estimate-the-standard-error-of-the-mle?rq=1

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