A note on the renormalization group approach to the Central Limit Theorem

Two proofs of the Central Limit Theorem using a renormalization group approach are presented. The first proof is conducted under a third moment assumption and shows that a suitable renormalization group map is a contraction over the space of probability measures with a third moment. The second proof uses Lyapunov stability and works under a second moment condition. These are by far not the most optimal proofs of the CLT, and the main interest of the proofs are their existence, the CLT being the simplest case in which a renormalization group argument should apply. None of the tools used in this note are new. Similar proofs are known amongst expert in limit theorems, but explicit references are not so easy to come by for non-experts in the field.