Open Access
February 2019 Sum rules and large deviations for spectral matrix measures
Fabrice Gamboa, Jan Nagel, Alain Rouault
Bernoulli 25(1): 712-741 (February 2019). DOI: 10.3150/17-BEJ1003


In the paradigm of random matrices, one of the most classical object under study is the empirical spectral distribution. This random measure is the uniform distribution supported by the eigenvalues of the random matrix. In this paper, we give large deviation theorems for another popular object built on Hermitian random matrices: the spectral measure. This last probability measure is a random weighted version of the empirical spectral distribution. The weights involve the eigenvectors of the random matrix. We have previously studied the large deviations of the spectral measure in the case of scalar weights. Here, we will focus on matrix valued weights. Our probabilistic results lead to deterministic ones called “sum rules” in spectral theory. A sum rule relative to a reference measure on $\mathbb{R}$ is a relationship between the reversed Kullback–Leibler divergence of a positive measure on $\mathbb{R}$ and some non-linear functional built on spectral elements related to this measure. By using only probabilistic tools of large deviations, we extend the sum rules to the case of Hermitian matrix-valued measures.


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Fabrice Gamboa. Jan Nagel. Alain Rouault. "Sum rules and large deviations for spectral matrix measures." Bernoulli 25 (1) 712 - 741, February 2019.


Received: 1 February 2017; Revised: 1 November 2017; Published: February 2019
First available in Project Euclid: 12 December 2018

zbMATH: 07007222
MathSciNet: MR3892334
Digital Object Identifier: 10.3150/17-BEJ1003

Keywords: large deviations , matrix-valued measures , orthogonal matrix polynomials , random matrices , sum rules

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 1 • February 2019
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