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Cohérence de grandes matrices aléatoires. Théorèmes limites et applications

Abstract : This thesis focuses on the study of the τ -coherence of an high-dimensional (n x p)-observation matrix with p >> n, where n is the number of individuals and p the number of variables. The τ -coherence is defined as the largest magnitude of the entries of the empirical correlation matrix outside a central band (with a bandwith τ). The first chapter is devoted to the presentation of the Chen-Stein method, which is an approximation of weakly dependent events by a Poisson distribution, and to some bibliography concerning coherence. The second and third chapter focus on the limiting behaviour of τ -coherence in a case where the observations are assumed to be Gaussian with bandwise (resp. blockwise) covariance in Chapter 2 (resp. Chapter 3). In the last chapter, we propose a Monte-Carlo simulation procedure allowing us to study numerically the limiting distribution of the τ -coherence for large (Big Data) matrices. We use a splitting strategy of our matrices and HPC method such as GPGPU computation in order to, from one side, being able to compute correlation matrices even if they are too large to be loaded in a computer, and on the other side, to reduce computation time. Finally the appendix is devoted to some technical results.
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https://tel.archives-ouvertes.fr/tel-03642453
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Submitted on : Friday, April 15, 2022 - 10:01:09 AM
Last modification on : Thursday, October 20, 2022 - 3:54:52 AM

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Maxime Boucher. Cohérence de grandes matrices aléatoires. Théorèmes limites et applications. Mathématiques générales [math.GM]. Université d'Orléans, 2021. Français. ⟨NNT : 2021ORLE3133⟩. ⟨tel-03642453⟩

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