Sum rules via large deviations

Fabrice Gamboa (1, 2) , Jan Nagel (3) , Alain Rouault (4)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 GdR MASCOT-NUM - Méthodes d'Analyse Stochastique des Codes et Traitements Numériques
3 TUM - Technische Universität Munchen - Technical University Munich - Université Technique de Munich
4 LMV - Laboratoire de Mathématiques de Versailles

In the theory of orthogonal polynomials, sum rules are remarkable relationships between a functional defined on a subset of all probability measures involving the reverse Kullback-Leibler divergence with respect to a particular distribution and recursion coefficients related to the orthogonal polynomial construction. Killip and Simon (Killip and Simon (2003)) have given a revival interest to this subject by showing a quite surprising sum rule for measures dominating the semicircular distribution on [−2, 2]. This sum rule includes a contribution of the atomic part of the measure away from [−2, 2]. In this paper, we recover this sum rule by using probabilistic tools on random matrices. Furthermore, we obtain new (up to our knowledge) magic sum rules for the reverse Kullback-Leibler divergence with respect to the Marchenko-Pastur or Kesten-McKay distributions. As in the semicircular case, these formulas include a contribution of the atomic part appearing away from the support of the reference measure.

Mots clés: Kullback-Leibler divergence; Jacobi matrix; Sum rules; large deviations; spectral measures; orthogonal polyno-mials; random matrices

Domaines: Mathématiques [math]; Statistiques [math.ST]

Fabrice Gamboa, Jan Nagel, Alain Rouault. Sum rules via large deviations. Journal of Functional Analysis, 2016, 270 (2), pp.509-559. ⟨10.1016/j.jfa.2015.08.009⟩. ⟨hal-01168243⟩