The central limit theorem via doubling of variables

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 $\mathbb{R}^d$.