Here’s a very interesting story about how the generalized central limit theorem of statistics can be derived from two complementary notions of the renormalization group:
The references where I found a useful exposition of this topic were the book (Amir 2020) and this very nice paper: (Calvo et al. 2010).
References
Amir, Ariel. 2020. Thinking Probabilistically: Stochastic Processes, Disordered Systems, and Their Applications. Cambridge University Press. https://doi.org/10.1017/9781108855259.
Calvo, Iván, Juan C. Cuchı́, José G. Esteve, and Fernando Falceto. 2010. “Generalized Central Limit Theorem and Renormalization Group.” Journal of Statistical Physics 141 (3): 409–21. https://doi.org/10.1007/s10955-010-0065-y.