Sum rules and large deviations for spectral matrix measures

Fabrice Gamboa (1) , Jan Nagel (2) , Alain Rouault (3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 TUM - Technische Universität Munchen - Technical University Munich - Université Technique de Munich
3 LMV - Laboratoire de Mathématiques de Versailles

A sum rule is a relationship between the reversed Kullback-Leibler divergence of a positive measure on R and some non-linear functional built on spectral elements related to this measure (see for example [13] or [20]). In this paper, using only probabilistic tools of large deviations, we extend the sum rules obtained in [9] to the case of an Hermitian matrix valued measure. We recover earlier result of Damanik et al ([3]) for the case of the semi-circular law and obtain new sum rules for Hermitian matrix measures in the Pastur-Marchenko case.

Domaines: Probabilités [math.PR]; Analyse classique [math.CA]

Fabrice Gamboa, Jan Nagel, Alain Rouault. Sum rules and large deviations for spectral matrix measures. 2016. ⟨hal-01264137v1⟩