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

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