Fluctuations of eigenvalues and second order Poincar\'e inequalities

TL;DR

This paper introduces second order Poincaré inequalities as a unified, softer approach to deriving Gaussian central limit theorems for eigenvalue statistics in random matrices, simplifying proofs and extending results.

Contribution

It formulates second order Poincaré inequalities, providing a unified framework for CLTs of eigenvalues, and applies Stein's method to establish these results with new examples like Gaussian Toeplitz matrices.

Findings

Established CLTs for eigenvalue statistics using second order Poincaré inequalities.

Provided a unified, less complex proof technique applicable to various random matrix models.

Derived a new CLT for the spectrum of Gaussian Toeplitz matrices.

Abstract

Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments. In the process, we introduce a notion of `second order Poincar\'e inequalities': just as ordinary Poincar\'e inequalities give variance bounds, second order Poincar\'e inequalities give central limit theorems. The proof of the main result employs Stein's method of normal approximation. A number of examples are worked out, some of which are new. One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices.