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Produits de matrices aléatoires :exposants de Lyapunov pour des matrices aléatoires suivant une mesure de Gibbs, théorèmes limites pour des produits au sens max-plus

Abstract : A stochastic recurrent sequences (SRS) driven by a sequence of random matrices is a sequence of random variables whose element of order n+1 is the product of the element of order n by the n-th matrix. This thesis investigates the asymptotic behaviour of such sequences. In the first part, matrices are invertible and we give a separation criterion for Lyapunov exponents when the sequence of matrices obeys to a Gibbs measure on a finite type subshift. In the second part, the products are in the max-plus algebra. We show that the first order behaviour of the SRS essentially depends on the behaviour of diagonal blocks and that the memory loss property, which ensures the stability of the SRS, is generic. Using the spectral gap method, we show that if the sequence of random matrices (or topical applications) is i.i.d. and has the memory loss property, then the SRS driven by this sequence satisfy classical limit theorems.
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https://tel.archives-ouvertes.fr/tel-00010813
Contributor : Glenn Merlet <>
Submitted on : Friday, October 28, 2005 - 11:02:37 AM
Last modification on : Friday, July 10, 2020 - 4:04:58 PM
Long-term archiving on: : Friday, April 2, 2010 - 11:06:11 PM

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  • HAL Id : tel-00010813, version 1

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Glenn Merlet. Produits de matrices aléatoires :exposants de Lyapunov pour des matrices aléatoires suivant une mesure de Gibbs, théorèmes limites pour des produits au sens max-plus. Mathématiques [math]. Université Rennes 1, 2005. Français. ⟨tel-00010813⟩

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