A new Berry-Esseen-type estimate in the free central limit theorem

TL;DR

This work advances rates of convergence in the free central limit theorem by establishing a Berry-Esseen-type bound in the Kolmogorov distance to the Wigner semicircle, expressed via the fourth Lyapunov fraction $L_{4n}$. Leveraging the subordination representation for free additive convolutions and Bai's smoothing inequality, the authors prove that, for any $\varepsilon\in(0,\tfrac{1}{2})$, $\Delta(\mu_{S_n},\omega) \le C_\varepsilon L_{4n}^{\tfrac{1}{2}-\varepsilon}$, with a recursive improvement mechanism that sharpens the rate when $L_{4n}$ is small. In the i.i.d. case, $L_{4n}$ scales as $n^{-1}$, yielding almost optimal rates $\Delta(\mu_{S_n},\omega) \le C_\varepsilon n^{-\tfrac{1}{2}+\varepsilon}$. The results illuminate when the new bound improves upon previous Lyapunov-based rates and extend Berry-Esseen-type convergence to non-identically distributed free sums, broadening the toolkit for quantitative free probability.

Abstract

Using the subordination approach, we provide a new Berry-Esseen-type estimate in the free central limit theorem in terms of the fourth Lyapunov fraction. In the special case of identical distributions, our result implies a rate of order $n^{-1/2 + \varepsilon}$ for any $\varepsilon>0$, thus almost leading to the optimal rate of order $n^{-1/2}$.

Theorems & Definitions (6)

• Theorem 1.1
• Theorem 2.1
• Theorem 2.2
• Theorem 3.1
• Corollary 3.2
• proof : Proof of \ref{['main theorem']}