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

• 成果类型: Article
• 署名作者: Neufeld, Leonie
• 署名单位: University of Bielefeld
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01294-0
• 发表日期: 2024
• 页码: 803-879
• 关键词: central limit; superconvergence; distributions; CONVERGENCE

摘要

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(-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.