论文标题
关于二阶非交通概率空间的分析结构以及有限的fréchet变化的功能
On the Analytic Structure of Second-Order Non-Commutative Probability Spaces and Functions of Bounded Fréchet Variation
论文作者
论文摘要
在本文中，我们基于有界féchet变化的函数，提出了一种新方法，以实现依赖于：操作员规范的大偏差原理的较弱形式的随机矩阵集合的连续线性线性统计量的函数；线性统计数据的庞加莱型不平等；以及二阶限制分布的存在。这种方法将许多已知的随机矩阵集合框架成单个设置，因此，恢复了线性统计的经典中心限制定理，并确定了新的统计量，例如，块块高斯矩阵的连续不同线性统计量的CLT。 此外，我们的主要结果有助于理解二阶非交通概率空间的分析结构。一方面，他们指出了与这些空间相关的双线性功能的无限性质的来源。另一方面，它们导致了二阶Cauchy Transform的整体表示形式的一般原型，$ g_2 $。此外，我们确定分解的协方差会融合到此转换中，并且分析线性统计的限制协方差可以表示为$ g_2 $中的轮廓积分。
In this paper we propose a new approach to the central limit theorem (CLT), based on functions of bounded Féchet variation for the continuously differentiable linear statistics of random matrix ensembles which relies on: a weaker form of a large deviation principle for the operator norm; a Poincaré-type inequality for the linear statistics; and the existence of a second-order limit distribution. This approach frames into a single setting many known random matrix ensembles and, as a consequence, classical central limit theorems for linear statistics are recovered and new ones are established, e.g., the CLT for the continuously differentiable linear statistics of block Gaussian matrices. In addition, our main results contribute to the understanding of the analytical structure of second-order non-commutative probability spaces. On the one hand, they pinpoint the source of the unbounded nature of the bilinear functional associated to these spaces; on the other hand, they lead to a general archetype for the integral representation of the second-order Cauchy transform, $G_2$. Furthermore, we establish that the covariance of resolvents converges to this transform and that the limiting covariance of analytic linear statistics can be expressed as a contour integral in $G_2$.
