Weighted sums and Berry-Esseen type estimates in free probability theory

We study weighted sums of free identically distributed self-adjoint random variables with weights chosen randomly from the unit sphere and show that the Kolmogorov distance between the distribution of such a weighted sum and Wigner’s semicircle law is of order \(n^{-\frac{1}{2}}\) with high probability. Replacing the Kolmogorov distance by a weaker pseudometric, we obtain a rate of convergence of order \(n^{-1}\), thus providing a free analog of the Klartag-Sodin result in classical probability theory. Moreover, we show that our ideas generalize to the setting of sums of free non-identically distributed bounded self-adjoint random variables leading to a new rate of convergence in the free central limit theorem.