From classical to modern central limit theorems

TL;DR

This paper reviews the historical development of classical and martingale central limit theorems and introduces a new direction called nonlinear central limit theorem and nonlinear normal distribution.

Contribution

It provides a historical overview and introduces the concept of nonlinear central limit theorems, expanding the theoretical framework of distribution convergence.

Findings

Historical review of classical and martingale CLTs

Introduction of nonlinear CLT and nonlinear normal distribution

Potential new research directions in probability theory

Abstract

De Moivre (1733), investigating the limit distribution of the binomial distribution, was the first to discover the existence of the normal distribution and the central limit theorem. In this review article, we briefly recall the history of classical central limit theorem and martingale central limit theorem, and introduce a new direction of central limit theorem, namely nonlinear central limit theorem and nonlinear normal distribution.