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hal-00531668, version 1

## Large Deviations for Random Matricial Moment Problems

Jan Nagel, Jens Wagener, Fabrice Gamboa 1, Alain Rouault 2

Journal of Multivariate Analysis 106 (2012) 17-35

• 1:  Institut de Mathématiques de Toulouse (IMT)

• Université Paul Sabatier [UPS] - Toulouse III – Université Toulouse le Mirail - Toulouse II – Université des Sciences Sociales - Toulouse I – Institut National des Sciences Appliquées (INSA) - Toulouse – CNRS : UMR5219 Bâtiment 1R3 118 route de Narbonne 31062 TOULOUSE CEDEX 4 France
• 2:  Laboratoire de Mathématiques de Versailles (LM-Versailles)
• http://www.math.uvsq.fr
CNRS : UMR8100 – Université de Versailles Saint-Quentin-en-Yvelines 45, avenue des Etats-Unis 78035 VERSAILLES cedex France

### Bibliographic reference

• Type of document: Articles in peer-reviewed journal
• Subject: Mathematics/Probability
• Title: Large Deviations for Random Matricial Moment Problems
• Abstract: We consider the moment space $\mathcal{M}_n^{K}$ corresponding to $p \times p$ complex matrix measures defined on $K$ ($K=[0,1]$ or $K=\D$). We endow this set with the uniform law. We are mainly interested in large deviations principles (LDP) when $n \rightarrow \infty$. First we fix an integer $k$ and study the vector of the first $k$ components of a random element of $\mathcal{M}_n^{K}$. We obtain a LDP in the set of $k$-arrays of $p\times p$ matrices. Then we lift a random element of $\mathcal{M}_n^{K}$ into a random measure and prove a LDP at the level of random measures. We end with a LDP on Carathéodory and Schur random functions. These last functions are well connected to the above random measure. In all these problems, we take advantage of the so-called canonical moments technique by introducing new (matricial) random variables that are independent and have explicit distributions.
• Fulltext language: English
• Production date: 2010-11-01
• DOI: 10.1016/j.jmva.2011.11.006
• Journal:  Journal of Multivariate Analysis Publisher Elsevier ISSN 0047-259X (eISSN : 1095-7243)
• Audience: international
• Publication date: 2012-04
• Volume: 106
• Page, identifiant, ...: 17-35
• Keyword(s): Random matrices – moments spaces – canonical moments – large deviations – Carathéodory functions – Schur functions
• Classification: 15B52, 60F10
• Comment: 32 pages
• ANR Project:  Project Id Grandes Matrices Aléatoires ANR-08-BLAN-0311-01

• hal-00531668, version 1
• oai:hal.archives-ouvertes.fr:hal-00531668
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• Submitted on: Wednesday, 3 November 2010 14:40:11
• Updated on: Wednesday, 27 February 2013 11:22:34