It is well known that, under standard regularity conditions, the maximum likelihood estimator (MLE) satisfies a central limit theorem and converges in distribution to a Gaussian random variable as the sample size grows. This paper strengthens this classical result by developing several stronger forms of asymptotic normality for the normalized MLE. With additional assumptions on the score, we first establish sub-Gaussian tail bounds and convergence of all moments for the normalized estimation error. We then prove an entropic central limit theorem for a smoothed version of the estimator, showing convergence in relative entropy to the limiting Gaussian law. When the Fisher information of the normalized estimate is bounded, or its density has bounded first derivative, we further show that the smoothing can be removed, yielding entropic normality of the MLE itself. The proofs develop auxiliary tools that may be of independent interest, including exponential consistency bounds, high-moment estimates, and entropy-control arguments for the estimator.

翻译：众所周知，在标准正则性条件下，极大似然估计量满足中心极限定理，并随着样本量增长依分布收敛于高斯随机变量。本文通过发展规范化极大似然估计量的若干更强形式的渐近正态性，强化了这一经典结论。在得分函数附加假设下，我们首先建立了规范化估计误差的次高斯尾界与各阶矩收敛性。随后证明了该估计量平滑版本的熵中心极限定理，揭示其相对熵收敛于极限高斯分布。当规范化估计的Fisher信息有界或其密度一阶导数有界时，进一步证明可去除平滑化，得到极大似然估计量本身的熵正态性。证明过程中发展了一系列可能具有独立价值的辅助工具，包括指数相合性界、高阶矩估计以及估计量的熵控制论证。