AI 中文总结
本文利用单位圆上的正交多项式理论和Stein方法，直接证明了圆β-系综幂的迹的高斯极限，并借助Verblunsky系数的显式耦合获得了几乎必然和$L^2$收敛。
AI 中文摘要
考虑圆β-系综，Jiang和Matsumoto计算了迹的矩，并利用Jack函数证明了多项式线性统计量的中心极限定理。我们利用单位圆上的正交多项式理论和Stein方法，直接证明了圆β-系综幂的迹的高斯极限。证明依赖于圆β-系综的Verblunsky系数的显式耦合，这也使我们能够获得几乎必然收敛和$L^2$收敛。
英文摘要
Consider the circular beta-ensembles, Jiang and Matsumoto computed the moments of traces, and proved central limit theorems for polynomial linear statistics using Jack functions. We give a direct proof of the Gaussian limits for the traces of powers of circular beta-ensembles using the theory of orthogonal polynomials on the unit circle and Stein's method. The proof relies on an explicit coupling of the Verblunsky coefficients of circular beta-ensembles, which also allow us to obtain almost sure and $L^2$ convergence.
Comments22 pages, no figures, comments are welcome
