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Théorèmes de renouvellement pour des fonctionnelles additives associées à des chaînes de Markov fortement ergodiques

Abstract : The purpose of this work is to extend the renewal theorems of the independent case to Markov additive functionals. Specifically, this thesis generalizes the one-dimensional results obtained by Yves Guivarc'h and the multidimensional ones established by Martine Babillot. As in these earlier works, the driving Markov chain is assumed to be strongly ergodic. The proofs are based on the Nagaev-Guivarc'h method involving both Fourier techniques and operator perturbation theory. The Fourier analysis (Chapter 2) borrows Babillot's method, but here the distribution-type arguments and the use of modified Bessel functions are replaced by some more elementary computations. The material of functional analysis is presented in Chapter 3. In Chapter 4, the Markov renewal theorems of Martine Babillot and Yves Guivarc'h are deduced from the results of the two previous chapters. In the Chapters 5 and 6, the spectral method is applied using the perturbation theorem due to Keller and Liverani in place of the standard perturbation theory. This new approach, inspired by the recent works of Hubert Hennion, Loïc Hervé and Françoise Pène, allows to significantly improve the Markov renewal theorems in terms of moment conditions. In particular, for the following models - the V-geometrical ergodic Markov chains, - the rho-mixing Markov chains, - the Lipschitz iterative models, the assumptions reduce to (almost) optimal moment conditions (with respect to the independent case). The applications to Lipschitz iterative models (Chapter 6) focus on the additive functionals associated with a double Markov chain involving the model and the underlying random Lipschitz maps. The results of this chapter are obtained by extending the definition of the weighted Lipschitz-type spaces introduced by Emile Le Page.
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Submitted on : Tuesday, April 5, 2011 - 9:43:33 AM
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Denis Guibourg. Théorèmes de renouvellement pour des fonctionnelles additives associées à des chaînes de Markov fortement ergodiques. Mathématiques [math]. Université Rennes 1, 2011. Français. ⟨tel-00583175⟩

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