Deriving the Central Limit Theorem from the De Moivre-Laplace Theorem

TL;DR

This paper presents a proof of the central limit theorem for any finite-variance random variable derived directly from the de Moivre-Laplace theorem, avoiding advanced mathematical tools.

Contribution

It provides a novel, elementary proof of the central limit theorem based solely on the de Moivre-Laplace theorem, without using characteristic functions or stochastic processes.

Findings

Central limit theorem extended to all finite-variance variables

Elementary proof avoiding advanced concepts

Direct derivation from de Moivre-Laplace theorem

Abstract

The de Moivre-Laplace theorem is a special case of the central limit theorem for Bernoulli random variables, and can be proved by direct computation. We deduce the central limit theorem for any random variable with finite variance from the de Moivre-Laplace theorem. Our proof does not use advanced notions such as characteristic functions, the Brownian motion, or stopping times.