On the Analytic Structure of Second-Order Non-Commutative Probability Spaces and Functions of Bounded Fr\'echet Variation

TL;DR

This paper introduces a new approach to the central limit theorem for random matrix linear statistics using functions of bounded Fréchet variation, and explores the analytical structure of second-order non-commutative probability spaces.

Contribution

It develops a unified framework for various random matrix ensembles, establishes new CLTs, and elucidates the structure of second-order non-commutative probability spaces.

Findings

Recovery of classical CLTs for linear statistics

Establishment of CLT for block Gaussian matrices

Insight into the unbounded nature of the bilinear functional

Abstract

In this paper we propose a new approach to the central limit theorem (CLT), based on functions of bounded F\'echet 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 Poincar\'{e}-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 $G_2$.