Renormalization ideas have spread across many mathematical disciplines since their introduction in quantum electrodynamics. While manifestations of renormalization across physics and dynamical systems appear similar, direct connections between, say, the block-spin renormalization of statistical physics and the return map renormalization of dynamical systems are often mysterious or absent. In this talk, I will present a renormalization group argument for local Berry-Esseen type bounds for normalized sums of i.i.d. random variables. This renormalization operator will be emblematic of a more general real-space block-spin renormalization operator for Dyson hierarchical spin models. By way of a Brydges-Fröhlich-Spencer random walk model representation for spin models, I will connect this to a renormalization procedure for random walks on certain self-similar groups. In this context, block-spin renormalization will be shown to be distinct from the return map renormalization (Schur renormalization) defined by Bartholdi, Kaimanovich, and Virag.