Unlock: Central Limit Theorem

The CLT family: classical Lindeberg-Levy, the Lindeberg-Feller condition for non-identical summands, Berry-Esseen rate, multivariate CLT, the delta method, and the martingale CLT that underwrites SGD asymptotics. Why these results power MLE asymptotic normality, bootstrap validity, and the Gaussian-noise approximation of stochastic optimization, and where the CLT breaks in high dimensions and under heavy tails.