A fixed point approach to non-commutative central limit theorems

Abstract We show how the renormalization group approach can be used to prove quantitative central limit theorems (CLTs) in the setting of free, Boolean, bi-free, and bi-Boolean independence under finite third moment assumptions. The proofs rely on the construction of a contraction on a subspace of probability measures over $$\mathbb {R}$$ R (or $$\mathbb {R}^2$$ R 2 ) equipped with a suitable metric, which has the appropriate analogue of a Gaussian distribution as a fixed point (for instance, the semi-circle law in the case of free independence). In all cases, this yields a convergence rate of $$1/\sqrt{n}$$ 1 / n , and we show that this can be improved to 1/ n in some instances under stronger assumptions.