A proof of the central limit theorem using the -Wasserstein metric
Calvin Wooyoung Chin
Apple Inc., Cupertino, USA

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 -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.
Cite this article
Calvin Wooyoung Chin, A proof of the central limit theorem using the -Wasserstein metric. Enseign. Math. (2025), published online first
DOI 10.4171/LEM/1094