A Proof of the Central Limit Theorem Using the $2$-Wasserstein Metric

TL;DR

This paper provides a novel proof of the central limit theorem by employing the 2-Wasserstein metric to measure distribution proximity to normality, avoiding traditional characteristic functions and Taylor expansions.

Contribution

It introduces a new proof technique for the CLT using the 2-Wasserstein metric and renormalization group methods, with detailed explanations and illustrations.

Findings

Proof of the CLT using Wasserstein metric

Avoids characteristic functions and Taylor expansions

Provides detailed metric properties and illustrations

Abstract

We prove the Lindeberg--Feller central limit theorem without using characteristic functions or Taylor expansions, but instead by measuring how far a distribution is from the standard normal distribution according to the $2$-Wasserstein metric. This falls under the category of renormalization group methods. The facts we need about the metric are explained and proved in detail. We illustrate the idea on a classical version of the central limit theorem before going into the main proof.