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Theses

Principe conditionnel de Gibbs pour des contraintes fines approchées et Inégalités de transport

Abstract : This thesis deals with two different subjects : Conditional principle of Gibbs type and transportation cost inequalities. In the first part of our work, we study the asymptotic behaviour of random measures, satisfying a large deviations principle, knowing that a rare event has occurred. Our aim is to study the case where the set defining the conditioning event is of probability zero. Our strategy is to progressively approximate this thin set by a sequence of larger sets. This approach, which requires exact controls of small probabilities, enables us to give a simple limit formulation of different conditional principles. The second part deals with transportation cost inequalities : one wants to majorize an optimal transportation cost by a concave function of relative entropy. Our goal is to put in light the links between these inequalities and Large Deviations theory. We show that transportation cost inequalities admit a dual formulation involving the Laplace-transform. Thanks to this property, we prove a general tensorization formula for transportation cost inequalities, which in turn yields deviation bounds for empirical processes. We establish also necessary and sufficient conditions for a large class of transportation inequalities.
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Submitted on : Friday, September 16, 2005 - 2:41:04 PM
Last modification on : Tuesday, March 2, 2021 - 10:02:46 AM
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  • HAL Id : tel-00010173, version 1

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Nathaël Gozlan. Principe conditionnel de Gibbs pour des contraintes fines approchées et Inégalités de transport. Mathématiques [math]. Université de Nanterre - Paris X, 2005. Français. ⟨tel-00010173⟩

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