Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness / Hubert Hennion, Loïc Hervé

Auteur: Hennion, Hubert (1944-) - AuteurCo-auteur: Hervé, Loïc (1963-) - AuteurType de document: Livre numériqueCollection: Lecture notes in mathematics, (Online) ; 1766Langue: anglaisÉditeur: Berlin : Springer, 2001 ISBN: 9783540424154 ISSN: 1617-9692Note: The usefulness of techniques from the perturbation theory of operators, applied to a quasi-compact positive kernel Q, for obtaining limit theorems for Markov chains or for describing stochastic properties of dynamical systems, by use of a Perron-Frobenius operator, has been demonstrated in several papers. The general features of all these works are the same. Particularities of implementation depend mainly on the functional spaces on which the quasi-compactness of Q has been proved and on the number of eigenvalues of the greatest modulus of Q. In this work the authors give a general functional framework for this method and establish central limit type theorems, large deviation theorems and renewal theorems. When applied to Lipschitz Markov kernels or to Perron-Frobenius operators associated with expanding maps, these statements give rise to new results and clarify the proofs of already known properties. The main part of this work deals with a quasi-compact Markov kernel Q for which 1 is a simple eigenvalue of modulus 1. In the last part the results of the preceding chapters are extended by the second author to quasi-compact kernels for which 1 is an eigenvalue of multiplicity greater than one. All chapters begin with a short presentation. (MathSciNet) Sujets MSC: 60J10 Probability theory and stochastic processes -- Markov processes -- Markov chains (discrete-time Markov processes on discrete state spaces)
60F10 Probability theory and stochastic processes -- Limit theorems -- Large deviations
37A25 Dynamical systems and ergodic theory -- Ergodic theory -- Ergodicity, mixing, rates of mixing
37A50 Dynamical systems and ergodic theory -- Ergodic theory -- Relations with probability theory and stochastic processes
60F05 Probability theory and stochastic processes -- Limit theorems -- Central limit and other weak theorems
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The usefulness of techniques from the perturbation theory of operators, applied to a quasi-compact positive kernel Q, for obtaining limit theorems for Markov chains or for describing stochastic properties of dynamical systems, by use of a Perron-Frobenius operator, has been demonstrated in several papers. The general features of all these works are the same. Particularities of implementation depend mainly on the functional spaces on which the quasi-compactness of Q has been proved and on the number of eigenvalues of the greatest modulus of Q. In this work the authors give a general functional framework for this method and establish central limit type theorems, large deviation theorems and renewal theorems. When applied to Lipschitz Markov kernels or to Perron-Frobenius operators associated with expanding maps, these statements give rise to new results and clarify the proofs of already known properties. The main part of this work deals with a quasi-compact Markov kernel Q for which 1 is a simple eigenvalue of modulus 1.
In the last part the results of the preceding chapters are extended by the second author to quasi-compact kernels for which 1 is an eigenvalue of multiplicity greater than one.
All chapters begin with a short presentation. (MathSciNet)

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