Weighted Sums and Berry-Esseen type estimates in Free Probability Theory

Leonie Neufeld

Published 2023-03-11, updated 2023-05-03Version 2

We study weighted sums of free identically distributed 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 $(\log n)^{\frac{1}{2}}n^{-\frac{1}{2}}$ with high probability. In the special case of bounded random variables the rate can be improved to $n^{-\frac{1}{2}}$. Replacing the Kolmogorov distance by a weaker pseudometric, we obtain a rate of convergence of order $(\log n)n^{-1}$. Our results can be seen as a free analogue of the Klartag-Sodin result in classical probability theory. Moreover, we show that our ideas generalise to the setting of sums of free non-identically distributed bounded random variables providing a new rate of convergence in the free central limit theorem.