论文标题
动态系统的免费中央限制定理
A free central-limit theorem for dynamical systems
论文作者
论文摘要
自由概率的基本定理的自由中央限制定理指出，自由独立的随机变量的经验平均值是渐近的半圆形。我们将此定理扩展到运算符的一般动力学系统，我们使用免费的随机变量$ x $以及一组 *-automorphims来定义，描述了$ x $的演变。我们引入了免费的混合系数，该系数衡量了动态系统距离自由独立的程度。在这些系数的条件下，我们证明了自由中央限制定理也适用于这些过程并提供浆果 - 埃森边界。我们将其推广到三角阵列和U统计数据。最后，我们通过一系列示例与经典概率和随机矩阵理论进行连接。
The free central-limit theorem, a fundamental theorem in free probability, states that empirical averages of freely independent random variables are asymptotically semi-circular. We extend this theorem to general dynamical systems of operators that we define using a free random variable $X$ coupled with a group of *-automorphims describing the evolution of $X$. We introduce free mixing coefficients that measure how far a dynamical system is from being freely independent. Under conditions on those coefficients, we prove that the free central-limit theorem also holds for these processes and provide Berry-Essen bounds. We generalize this to triangular arrays and U-statistics. Finally we draw connections with classical probability and random matrix theory with a series of examples.
