## The Annals of Probability

### Quantitative normal approximation of linear statistics of $\beta$-ensembles

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 47, Number 5 (2019), 2619-2685.

Dates
Received: September 2017
Revised: July 2018
First available in Project Euclid: 22 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1571731432

Digital Object Identifier
doi:10.1214/18-AOP1314

Mathematical Reviews number (MathSciNet)
MR4021234

#### Citation

Lambert, Gaultier; Ledoux, Michel; Webb, Christian. Quantitative normal approximation of linear statistics of $\beta$-ensembles. Ann. Probab. 47 (2019), no. 5, 2619--2685. doi:10.1214/18-AOP1314. https://projecteuclid.org/euclid.aop/1571731432

#### References

• [1] Anderson, G. W., Guionnet, A. and Zeitouni, O. (2010). An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge Univ. Press, Cambridge.
• [2] Bakry, D., Gentil, I. and Ledoux, M. (2014). Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 348. Springer, Cham.
• [3] Bekerman, F., Figalli, A. and Guionnet, A. (2015). Transport maps for $\beta$-matrix models and universality. Comm. Math. Phys. 338 589–619.
• [4] Bekerman, F., Leblé, T. and Serfaty, S. (2017). CLT for fluctuations of $\beta$-ensembles with general potential. Preprint. Available at arXiv:1706.09663.
• [5] Borot, G. and Guionnet, A. (2013). Asymptotic expansion of $\beta$ matrix models in the one-cut regime. Comm. Math. Phys. 317 447–483.
• [6] Borot, G. and Guionnet, A. (2013). Asymptotic expansion of beta matrix models in the multi-cut regime. Preprint. Available at arXiv:1303.1045.
• [7] Bourgade, P., Erdös, L. and Yau, H.-T. (2014). Edge universality of beta ensembles. Comm. Math. Phys. 332 261–353.
• [8] Breuer, J. and Duits, M. (2017). Central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients. J. Amer. Math. Soc. 30 27–66.
• [9] Cabanal-Duvillard, T. (2001). Fluctuations de la loi empirique de grandes matrices aléatoires. Ann. Inst. Henri Poincaré Probab. Stat. 37 373–402.
• [10] Chatterjee, S. (2009). Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Related Fields 143 1–40.
• [11] Döbler, C. and Stolz, M. (2011). Stein’s method and the multivariate CLT for traces of powers on the classical compact groups. Electron. J. Probab. 16 2375–2405.
• [12] Döbler, C. and Stolz, M. (2014). A quantitative central limit theorem for linear statistics of random matrix eigenvalues. J. Theoret. Probab. 27 945–953.
• [13] Fulman, J. (2012). Stein’s method, heat kernel, and traces of powers of elements of compact Lie groups. Electron. J. Probab. 17 no. 66, 16.
• [14] Grafakos, L. (2008). Classical Fourier Analysis, 2nd ed. Graduate Texts in Mathematics 249. Springer, New York.
• [15] Johansson, K. (1997). On random matrices from the compact classical groups. Ann. of Math. (2) 145 519–545.
• [16] Johansson, K. (1998). On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91 151–204.
• [17] Kriecherbauer, T. and Shcherbina, M. (2010). Fluctuations of eigenvalues of matrix models and their applications. Preprint. Available at arXiv:1003.6121.
• [18] Lambert, G. (2018). Limit theorems for biorthogonal ensembles and related combinatorial identities. Adv. Math. 329 590–648.
• [19] Leblé, T. and Serfaty, S. (2017). Large deviation principle for empirical fields of log and Riesz gases. Invent. Math. 210 645–757.
• [20] Ledoux, M. (2012). Chaos of a Markov operator and the fourth moment condition. Ann. Probab. 40 2439–2459.
• [21] Ledoux, M., Nourdin, I. and Peccati, G. (2015). Stein’s method, logarithmic Sobolev and transport inequalities. Geom. Funct. Anal. 25 256–306.
• [22] Ledoux, M. and Popescu, I. (2013). The one dimensional free Poincaré inequality. Trans. Amer. Math. Soc. 365 4811–4849.
• [23] Mason, J. C. and Handscomb, D. C. (2003). Chebyshev Polynomials. Chapman & Hall/CRC Press, Boca Raton, FL.
• [24] Meckes, E. (2009). On Stein’s method for multivariate normal approximation. In High Dimensional Probability V: The Luminy Volume. Inst. Math. Stat. (IMS) Collect. 5 153–178. IMS, Beachwood, OH.
• [25] Mehta, M. L. (2004). Random Matrices, 3rd ed. Pure and Applied Mathematics (Amsterdam) 142. Elsevier/Academic Press, Amsterdam.
• [26] Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus. From Stein’s Method to Universality. Cambridge Tracts in Mathematics 192. Cambridge Univ. Press, Cambridge.
• [27] Otto, F. and Villani, C. (2000). Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 361–400.
• [28] Pastur, L. (2006). Limiting laws of linear eigenvalue statistics for Hermitian matrix models. J. Math. Phys. 47 103303, 22.
• [29] Pastur, L. and Shcherbina, M. (2011). Eigenvalue Distribution of Large Random Matrices. Mathematical Surveys and Monographs 171. Amer. Math. Soc., Providence, RI.
• [30] Reed, M. and Simon, B. (1980). Methods of Modern Mathematical Physics. I. Functional Analysis, 2nd ed. Academic Press, New York.
• [31] Shcherbina, M. (2013). Fluctuations of linear eigenvalue statistics of $\beta$ matrix models in the multi-cut regime. J. Stat. Phys. 151 1004–1034.
• [32] Shcherbina, M. (2014). Change of variables as a method to study general $\beta$-models: Bulk universality. J. Math. Phys. 55 043504, 23.
• [33] Stein, C. (1995). The accuracy of the normal approximation to the distribution of the traces of powers of random orthogonal matrices. Technical Report 470, Dept. Statistics, Stanford Univ., Stanford, CA.
• [34] Titchmarsh, E. C. (1986). Introduction to the Theory of Fourier Integrals, 3rd ed. Chelsea, New York.
• [35] Tricomi, F. G. (1985). Integral Equations. Dover, New York.
• [36] Villani, C. (2009). Optimal Transport. Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer, Berlin.
• [37] Webb, C. (2016). Linear statistics of the circular $\beta$-ensemble, Stein’s method, and circular Dyson Brownian motion. Electron. J. Probab. 21 Paper No. 25, 16.