  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 contractive metric over the space of
probability measures over $\mathbb{R}$ or $\mathbb{R}^2$, 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 $1/\sqrt{n}$, and we show that this can be
improved to $1/n$ in some instances under stronger assumptions.
