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.
An Information-Theoretic Proof of the Central Limit Theorem with Lindeberg Conditions,
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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.