In this paper, we propose a new approach to the central limit theorem (CLT) based on functions of bounded Fréchet 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 $G_2$.

在本文中，我们提出了一种基于有界 Fréchet 变分函数的中心极限定理 (CLT) 的新方法，用于随机矩阵系综的连续可微线性统计，它依赖于算子范数的大偏差原理的较弱形式；线性统计的庞加莱型不等式；以及二阶极限分布的存在性。这种方法将许多已知的随机矩阵系综纳入单一设置，因此，恢复了线性统计的经典中心极限定理，并建立了新的中心极限定理，例如，用于分块高斯矩阵的连续可微线性统计的 CLT。此外，我们的主要结果有助于理解二阶非交换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泛函的无限性质的来源；另一方面，它们导致二阶柯西变换的积分表示的一般原型，$G_{\mathrm{2个}}$. 此外，我们确定分解的协方差收敛到这个变换，并且解析线性统计的极限协方差可以表示为等高线积分$G_{\mathrm{2个}}$.