AI 中文总结
本文针对自由独立自伴随机变量之和，证明了自由中心极限定理的Berry--Esseen估计，得到与经典概率论形式一致的界，改进了已有结果，证明结合截尾、R-变换估计等多种技术。
AI 中文摘要
我们研究未必同分布的自由独立自伴随机变量之和。设μ_j为第j个加项的分布，假设其均值为零且存在有限的2+δ阶绝对矩，其中0<δ≤1。设Δ为Kolmogorov距离，μ^(n)为规范化部分和的分布，ω为标准半圆律，B_n²为部分和的方差。本文旨在证明自由中心极限定理中的Berry--Esseen估计，即存在绝对常数C>0，使得对任意0<δ≤1，有Δ(μ^(n),ω) ≤ (C/B_n^(2+δ))∑_(j=1)^n ∫_R |x|^(2+δ) μ_j(dx)。该结果不仅改进了若干已知的一般非同分布随机变量估计，还建立了与经典概率论中形式完全一致的Berry--Esseen估计。证明结合了截尾、R-变换的定量估计、扰动半圆方程的稳定性分析以及Bai型光滑不等式。
英文摘要
We consider sums of freely independent self-adjoint random variables that are not necessarily identically distributed. Let $μ_j$ denote the distribution of the $j$th summand. We assume that they have mean zero and finite absolute moments of order $2+δ$, where $0<δ\le 1$. Let $Δ$ denote the Kolmogorov distance, let $μ^{(n)}$ be the distribution of the normalized partial sum, let $ω$ 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 $0<δ\le 1$, \[ Δ(μ^{(n)},ω) \le \frac{C}{B_n^{2+δ}}\sum_{j=1}^n \int_{\R}|x|^{2+δ}\,μ_j(dx), \] 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.
Comments17 pages, no figure
