  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.
