  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 $\mathbb{R}^d$.
