The Berry--Esseen Estimate in the Free Central Limit Theorem
We consider sums of freely independent self-adjoint random variables that are not necessarily identically distributed. Let $\mu_j$ denote the distribution of the $j$th summand. We assume that they have mean zero and finite absolute moments of order $2+\delta$, where $0<\delta\le 1$. Let $\Delta$ denote the Kolmogorov distance, let $\mu^{(n)}$ be the distribution of the normalized partial sum, let $\omega$ be the standard semicircle law, and let $B_n^2$ be the variance of the partial sum. The purpose of this paper is to prove the Berry--Esseen estimate in the free central limit theorem. Namely, there exists an absolute constant $C>0$ such that, for every $0<\delta\le 1$, \[ \Delta(\mu^{(n)},\omega) \le \frac{C}{B_n^{2+\delta}}\sum_{j=1}^n \int_{\R}|x|^{2+\delta}\,\mu_j(dx), \] Our result not only improves several known estimates for general non-identically distributed random variables, but also establishes exactly the same Berry--Esseen estimate as in classical probability theory. The proof combines truncation, a quantitative estimate for the $R$-transform, a stability analysis of a perturbed semicircle equation, and a Bai-type smoothing inequality.