Under regularity conditions plus assumptions on the score, the normalized MLE has sub-Gaussian tails, all moments converge, and the estimator converges in relative entropy to Gaussian when Fisher information is bounded or the density has bounded derivative.
Tests of statistical hypotheses concerning several parameters when the number of observations is large,
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Sub-Gaussian Concentration and Entropic Normality of the Maximum Likelihood Estimator
Under regularity conditions plus assumptions on the score, the normalized MLE has sub-Gaussian tails, all moments converge, and the estimator converges in relative entropy to Gaussian when Fisher information is bounded or the density has bounded derivative.