A New Direct Proof of the Central Limit Theorem

TL;DR

This paper presents a novel direct proof of the Central Limit Theorem utilizing Haar wavelet basis, calculus, and elementary probability, emphasizing the role of $L^{2}([0,1])$ and finite variance, while also estimating convergence rates.

Contribution

It introduces a new proof of the CLT based on weak convergence and Haar wavelets, highlighting the functional analysis perspective and providing convergence rate estimates.

Findings

Proof of CLT using Haar basis and elementary tools

Estimation of convergence rates and strong convergence away from tails

Clarification of the role of $L^{2}([0,1])$ in the CLT

Abstract

We prove the Central Limit Theorem (CLT) from the definition of weak convergence using the Haar wavelet basis, calculus, and elementary probability. The use of the Haar basis pinpoints the role of $L^{2}([0,1])$ in the CLT as well as the assumption of finite variance. We estimate the rate of convergence and prove strong convergence away from the tails.