A MODERN PERSPECTIVE ON THE CENTRAL LIMIT THEOREM: FROM CLASSICAL FOUNDATIONS TO HIGH DIMENSIONAL EXTENSIONS

Abstract

The Central Limit Theorem (CLT) stands as one of the cornerstones of probability theory and statistical inference. This thesis revisits the classical formulations—Lindeberg–Lévy, Lyapunov, and Lindeberg–Feller—and then explores two contemporary frontiers:  high‑dimensional CLTs where the dimension grows with sample size, and non‑identically distributed dependent structures arising in modern data science. The thesis presents a unified framework based on Stein’s method and Malliavin calculus that yields explicit rates of convergence (Berry–Esseen bounds) even in complex settings. Applications to hypothesis testing, bootstrap validity, and machine learning are discussed. The results provide theoretical guarantees that remain sharp under minimal moment assumptions, bridging classical probability with the demands of big data.