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2019 Convergence of the empirical spectral distribution of Gaussian matrix-valued processes
Arturo Jaramillo, Juan Carlos Pardo, José Luis Pérez
Electron. J. Probab. 24: 1-22 (2019). DOI: 10.1214/18-EJP203

Abstract

For a given normalized Gaussian symmetric matrix-valued process $Y^{(n)}$, we consider the process of its eigenvalues $\{(\lambda _{1}^{(n)}(t),\dots , \lambda _{n}^{(n)}(t)); t\ge 0\}$ as well as its corresponding process of empirical spectral measures $\mu ^{(n)}=(\mu _{t}^{(n)}; t\geq 0)$. Under some mild conditions on the covariance function associated to $Y^{(n)}$, we prove that the process $\mu ^{(n)}$ converges in probability to a deterministic limit $\mu $, in the topology of uniform convergence over compact sets. We show that the process $\mu $ is characterized by its Cauchy transform, which is a rescaling of the solution of a Burgers’ equation. Our results extend those of Rogers and Shi [14] for the free Brownian motion and Pardo et al. [12] for the non-commutative fractional Brownian motion when $H>1/2$ whose arguments use strongly the non-collision of the eigenvalues. Our methodology does not require the latter property and in particular explains the remaining case of the non-commutative fractional Brownian motion for $H< 1/2$ which, up to our knowledge, was unknown.

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Arturo Jaramillo. Juan Carlos Pardo. José Luis Pérez. "Convergence of the empirical spectral distribution of Gaussian matrix-valued processes." Electron. J. Probab. 24 1 - 22, 2019. https://doi.org/10.1214/18-EJP203

Information

Received: 27 January 2018; Accepted: 21 July 2018; Published: 2019
First available in Project Euclid: 16 February 2019

zbMATH: 07055648
MathSciNet: MR3916330
Digital Object Identifier: 10.1214/18-EJP203

Subjects:
Primary: 15B52, 46L54, 60H07, 65C30

JOURNAL ARTICLE
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