The central limit theorem via doubling of variables

TL;DR

This paper presents a novel proof of the multidimensional central limit theorem using the doubling variables technique, also providing quantitative bounds similar to Berry--Esseen for variables with finite moments.

Contribution

It introduces a self-contained proof method for the CLT using PDE techniques and extends quantitative bounds to higher dimensions.

Findings

Provides a new proof of the multidimensional CLT

Yields quantitative bounds for variables with finite moments

Establishes a version of the Berry--Esseen theorem in multiple dimensions

Abstract

We give a new, self-contained proof of the multidimensional central limit theorem using the technique of ``doubling variables," which is traditionally used to prove uniqueness of solutions of partial differential equations (PDEs). Our technique also yields quantitative bounds for random variables with finite $2+\gamma$ moment for some $\gamma \in (0,1]$; when $\gamma=1$, this proves a version of the Berry--Esseen theorem in $\mathbb{R}^d$.