A fixed-point approach to non-commutative central limit theorems

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 $\mathbb{R}$ (or ${\mathbb{R}}^2$) 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 $1\mathrm{/}\sqrt{n}$, and we show that this can be improved to $1\mathrm{/}n$ in some instances under stronger assumptions.