February 2024 Multivariate normal approximation for traces of orthogonal and symplectic matrices
Klara Courteaut, Kurt Johansson
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 60(1): 312-342 (February 2024). DOI: 10.1214/22-AIHP1332


We show that the distance in total variation between TrU,12TrU2,,1mTrUm and a real Gaussian vector, where U is a Haar distributed orthogonal or symplectic matrix of size 2n or 2n+1, is bounded by Γ2nm+112 times a correction. The correction term is explicit and holds for all nm4, for m sufficiently large. For nm3 we obtain the bound nmcnm with an explicit constant c. Our method of proof is based on an identity of Toeplitz + Hankel determinants due to Basor and Ehrhardt, see (Oper. Matrices 3 (2009) 167–86), which is also used to compute the joint moments of the traces.

Nous montrons que la distance en variation totale entre TrU,12TrU2,,1mTrUm et un vecteur gaussien réel, où U est une matrice orthogonale ou symplectique distribuée selon la mesure de Haar et de taille 2n ou 2n+1, est bornée par Γ2nm+112 fois une correction. Cette correction est explicite et valable pour tout nm4, pour m suffisamment grand. Lorsque nm3 nous obtenons la borne nmcnmc est une constante explicite. Notre méthode de démonstration repose sur une identité de déterminants du type Toeplitz + Hankel due à Basor et Ehrhardt, voir (Oper. Matrices 3 (2009) 167–86), qui est aussi utilisée pour calculer les moments joints des traces.


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Klara Courteaut. Kurt Johansson. "Multivariate normal approximation for traces of orthogonal and symplectic matrices." Ann. Inst. H. Poincaré Probab. Statist. 60 (1) 312 - 342, February 2024. https://doi.org/10.1214/22-AIHP1332


Received: 3 May 2021; Revised: 7 September 2022; Accepted: 3 October 2022; Published: February 2024
First available in Project Euclid: 3 March 2024

MathSciNet: MR4718383
Digital Object Identifier: 10.1214/22-AIHP1332

Primary: 47B35 , 60B12 , 60B15 , 60B20

Keywords: Hankel determinants , Multivariate Gaussian approximation , Toeplitz determinants

Rights: Copyright © 2024 Association des Publications de l’Institut Henri Poincaré


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Vol.60 • No. 1 • February 2024
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