From Classical to Modern Nonlinear Central Limit Theorems

In 1733, de Moivre, investigating the limit distribution of the binomial distribution, was the first to discover the existence of the normal distribution and the central limit theorem (CLT). This review article aims to provide a brief history of classical CLT and martingale CLT while introducing newer directions of CLT, specifically focusing on Peng’s nonlinear CLT and Chen–Epstein’s nonlinear CLT, as well as the Chen–Epstein nonlinear normal distribution function. The analysis revisits fundamental results such as the de Moivre–Laplace theorem, the Lindeberg–Feller theorem, the Lévy theorem, and Hall’s theorem, establishing connections between classical CLT concerning independent random variables and martingale CLT dealing with dependent summands. Both theorems share a framework involving linear expectations and probability spaces.