## Bernoulli

• Bernoulli
• Volume 25, Number 1 (2019), 712-741.

### Sum rules and large deviations for spectral matrix measures

#### Abstract

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.

#### Article information

Source
Bernoulli, Volume 25, Number 1 (2019), 712-741.

Dates
Revised: November 2017
First available in Project Euclid: 12 December 2018

https://projecteuclid.org/euclid.bj/1544605261

Digital Object Identifier
doi:10.3150/17-BEJ1003

Mathematical Reviews number (MathSciNet)
MR3892334

Zentralblatt MATH identifier
07007222

#### Citation

Gamboa, Fabrice; Nagel, Jan; Rouault, Alain. Sum rules and large deviations for spectral matrix measures. Bernoulli 25 (2019), no. 1, 712--741. doi:10.3150/17-BEJ1003. https://projecteuclid.org/euclid.bj/1544605261

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#### Supplemental materials

• Preliminaries on matrix measures. We refer to the supplementary file [13] for more details on matrix measures. It also explains how some of the results from the scalar case extend to the matrix case.