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Inégalités de déviations, principe de déviations modérées et théorèmes limites pour des processus indexés par un arbre binaire et pour des modèles markoviens

Abstract : The explicit control of the convergence of properly normalized sums of random variables, as well as the study of moderate deviation principle associated with these sums constitute the main subjects of this thesis. We mostly study two sort of processes. First, we are interested in processes labelled by binary tree, random or not. These processes have been introduced in the literature in order to study mechanism of the cell division. In Chapter 2, we study bifurcating Markov chains. These chains may be seen as an adaptation of "usual'' Markov chains in case the index set has a binary structure. Under uniform and non-uniform geometric ergodicity assumptions of an embedded Markov chain, we provide deviation inequalities and a moderate deviation principle for the bifurcating Markov chains. In chapter 3, we are interested in p-order bifurcating autoregressive processes (). These processes are an adaptation of p-order linear autoregressive processes in case the index set has a binary structure. We provide deviation inequalities, as well as an moderate deviation principle for the least squares estimators of autoregressive parameters of this model. In Chapter 4, we dealt with deviation deviation inequalities for bifurcating Markov chains on Galton-Watson tree. These chains are a generalization of the notion of bifurcating Markov chains in case the index set is a binary Galton-Watson tree. They allow, in case of cell division, to take into account cell's death. The main hypothesis that we do in this chapter are : uniform geometric ergodicity of an embedded Markov chain and the non-extinction of the associated Galton-Watson process. In Chapter 5, we are interested in first-order linear autoregressive models with correlated errors. More specifically, we focus on the Durbin-Watson statistic. The Durbin-Watson statistic is at the base of Durbin-Watson tests, which allow to detect serial correlation in the first-order autoregressive models. We provide a moderate deviation principle for this statistic. The proofs of moderate deviation principle of Chapter 2, 3 and 4 are essentially based on moderate deviation for martingales. To establish deviation inequalities, we use most the Azuma-Bennet-Hoeffding inequality and the binary structure of processes. Chapter 6 was born from the importance that explicit ergodicity of Markov chains has in Chapter 2. Since explicit geometric ergodicity of discrete and continuous time Markov processes has been well studied in the literature, we focused on the sub-exponential ergodicity of continuous time Markov Processes. We thus provide explicit rates for the sub-exponential convergence of a continuous time Markov process to its stationary distribution. The main hypothesis that we use are : existence of a Lyapunov fonction and of a minorization condition. The proofs are largely based on the coupling construction and the explicit control of the tail of the coupling time.
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https://tel.archives-ouvertes.fr/tel-00822136
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Siméon Valère Bitseki Penda. Inégalités de déviations, principe de déviations modérées et théorèmes limites pour des processus indexés par un arbre binaire et pour des modèles markoviens. Mathématiques générales [math.GM]. Université Blaise Pascal - Clermont-Ferrand II, 2012. Français. ⟨NNT : 2012CLF22291⟩. ⟨tel-00822136⟩

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