On the analytic structure of second-order non-commutative probability spaces and functions of bounded Fréchet variation

Abstract

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, [Formula: see text]. 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 [Formula: see text].