Fluctuations des marches aléatoires en dimension 1 Théorèmes limites locaux pour des marches réfléchies sur N

Abstract : The purpose of this thesis is to establish some local limit theorems for reflected random walks on N. The fluctuations theory and the Wiener-Hopf factorization play a crucial role. We will develop in the first part a classical approach that we will apply to the study of random walks on R+ with non-elastic reflections at zero. In the second part, we will explicit a different method which involves algebraic tools, complex analysis and factorization techniques, using in an essential way generating functions. These approach was developed 50 years ago to cover Markov walks, it will be presented in this part in the case of random walks with i.i.d jumps where many simplifications appear and will be then used to study random walks on N with either elastic or non-elastic reflections at zero. Finally, in the last part, we will introduce the useful tools to establish a Wiener-Hopf factorization in a markovian framework in order to study the fluctuations of Markov walks on Z. We investigate some previous work, especially some proofs that warranted to be more detailed, with a medium-term objective of applying the algebraic tools described above to study reflected Markov walks on N. Keywords : Markov chains, random walks, local limit theorems, fluctuations theory, Wiener-Hopf factorization, absorbed random walks, reflected random walks, Markov walks.
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Submitted on : Tuesday, June 10, 2014 - 6:00:58 PM
Last modification on : Thursday, January 17, 2019 - 3:02:01 PM
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Rim Essifi. Fluctuations des marches aléatoires en dimension 1 Théorèmes limites locaux pour des marches réfléchies sur N. Probabilités [math.PR]. Université François Rabelais - Tours, 2014. Français. ⟨tel-01003859⟩

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