November 2022 Fluctuations for matrix-valued Gaussian processes
Mario Diaz, Arturo Jaramillo, Juan Carlos Pardo
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 58(4): 2216-2249 (November 2022). DOI: 10.1214/21-AIHP1238


We consider a symmetric matrix-valued Gaussian process Y(n)=(Y(n)(t);t0) and its empirical spectral measure process μ(n)=(μt(n);t0). Under some mild conditions on the covariance function of Y(n), we find an explicit expression for the limit distribution of


where F=(f1,,fr), for r1, with each component belonging to a large class of test functions, and


More precisely, we establish the stable convergence of ZF(n) and determine its limiting distribution. An upper bound for the total variation distance of the law of Zf(n)(t) to its limiting distribution, for a test function f and t0 fixed, is also given.

Nous considérons un processus gaussien symétrique à valeurs matricielles Y(n)=(Y(n)(t);t0) et son processus des mesures spectrales empiriques μ(n)=(μt(n);t0). Sous des conditions assez faibles sur la fonction de covariance de Y(n) nous trouvons une expression explicite pour la loi limite de


F=(f1,,fr), pour r1, avec chaque composant appartenant à une grande classe des fonctions test, et


Plus précism´ent, nous établissons la convergence stable de ZF(n) et nous déterminons sa loi limite. Nous donnons également une borne supérieure pour la distance en variation totale entre la loi de Zf(n)(t) et sa loi limite, pour une fonction test f et t0 fixés.

Funding Statement

JCP acknowledges support from the Royal Society and CONACYT (CB-250590).
The work of MD was supported in part by the Consejo Nacional de Ciencia y Tecnología (CONACYT) under grant A1-S-976.


This work was concluded whilst JCP was on sabbatical leave holding a David Parkin Visiting Professorship at the University of Bath, he gratefully acknowledges the kind hospitality of the Department and University. This work was started when AJ was postdoctoral researcher jointly at the University of Luxembourg and the National University of Singapore.


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Mario Diaz. Arturo Jaramillo. Juan Carlos Pardo. "Fluctuations for matrix-valued Gaussian processes." Ann. Inst. H. Poincaré Probab. Statist. 58 (4) 2216 - 2249, November 2022.


Received: 10 February 2020; Revised: 8 December 2021; Accepted: 17 December 2021; Published: November 2022
First available in Project Euclid: 6 October 2022

MathSciNet: MR4492977
zbMATH: 1498.60030
Digital Object Identifier: 10.1214/21-AIHP1238

Primary: 60B20 , 60F05 , 60G15 , 60H05 , 60H07

Keywords: central limit theorem , Gaussian Orthogonal Ensemble , Malliavin calculus , Matrix-valued Gaussian processes , Skorokhod integration

Rights: Copyright © 2022 Association des Publications de l’Institut Henri Poincaré


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Vol.58 • No. 4 • November 2022
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