A Short and Elementary Proof of the Central Limit Theorem by Individual Swapping

TL;DR

This paper offers a concise, elementary proof of the central limit theorem using a simple swapping technique, avoiding advanced mathematical tools, and extends to the Lindeberg-Feller version.

Contribution

It introduces a straightforward proof method for the CLT based on individual swapping, making the proof accessible without advanced prerequisites.

Findings

Provides an elementary proof of the CLT

Extends proof to Lindeberg-Feller CLT

Avoids use of characteristic functions or linear operators

Abstract

We present a short proof of the central limit theorem which is elementary in the sense that no knowledge of characteristic functions, linear operators, or other advanced results are needed. Our proof is based on Lindeberg's trick of swapping a term for a normal random variable in turn. The modifications needed to prove the stronger Lindeberg-Feller central limit theorem are addressed at the end.