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September 2019 Quantitative normal approximation of linear statistics of $\beta$-ensembles
Gaultier Lambert, Michel Ledoux, Christian Webb
Ann. Probab. 47(5): 2619-2685 (September 2019). DOI: 10.1214/18-AOP1314


We present a new approach, inspired by Stein’s method, to prove a central limit theorem (CLT) for linear statistics of $\beta$-ensembles in the one-cut regime. Compared with the previous proofs, our result requires less regularity on the potential and provides a rate of convergence in the quadratic Kantorovich or Wasserstein-2 distance. The rate depends both on the regularity of the potential and the test functions, and we prove that it is optimal in the case of the Gaussian Unitary Ensemble (GUE) for certain polynomial test functions.

The method relies on a general normal approximation result of independent interest which is valid for a large class of Gibbs-type distributions. In the context of $\beta$-ensembles, this leads to a multi-dimensional CLT for a sequence of linear statistics which are approximate eigenfunctions of the infinitesimal generator of Dyson Brownian motion once the various error terms are controlled using the rigidity results of Bourgade, Erdős and Yau.


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Gaultier Lambert. Michel Ledoux. Christian Webb. "Quantitative normal approximation of linear statistics of $\beta$-ensembles." Ann. Probab. 47 (5) 2619 - 2685, September 2019.


Received: 1 September 2017; Revised: 1 July 2018; Published: September 2019
First available in Project Euclid: 22 October 2019

zbMATH: 07145300
MathSciNet: MR4021234
Digital Object Identifier: 10.1214/18-AOP1314

Primary: 60B20, 60F05, 60K35
Secondary: 82B05

Rights: Copyright © 2019 Institute of Mathematical Statistics


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Vol.47 • No. 5 • September 2019
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