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

By: Hennion, Hubert, 1944-
Contributor(s): Hervé, Loïc, 1963-
Material type: TextTextSeries: Lecture notes in mathematics (Springer-Verlag): 1766.Publisher: Berlin ; New York : Springer, ©2001Description: 1 online resource (144 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9783540446231; 3540446230; 3540424156; 9783540424154Subject(s): Markov processes | Limit theorems (Probability theory) | Differentiable dynamical systems | Stochastic processes | Differentiable dynamical systems | Limit theorems (Probability theory) | Markov processes | Stochastic processesGenre/Form: Electronic books. Additional physical formats: Print version:: Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness.DDC classification: 510 s 519.2/33 LOC classification: QA3 | .L28 no. 1766Online resources: Click here to access online
Contents:
General facts about the method, purpose of the paper -- The central limit theorems for Markov chains -- Quasi-compact operators of diagonal type and perturbations -- First properties of Fourier kernels, application -- Peripheral eigenvalues of Fourier kernels -- Proofs of theorems A, B, C -- Renewal theorem for Markov chains (theorem D) -- Large deviations for Markov chains (theorem E) -- Ergodic properties for Markov chains -- Stochastic properties of dynamical systems -- Expanding maps -- Proofs of some statements in probability theory -- Functional analysis results on quasi-compactness -- Generalization to the non-ergodic case (by L. Herv).
Summary: This book shows how techniques from the perturbation theory of operators, applied to a quasi-compact positive kernel, may be used to obtain limit theorems for Markov chains or to describe stochastic properties of dynamical systems. A general framework for this method is given and then applied to treat several specific cases. An essential element of this work is the description of the peripheral spectra of a quasi-compact Markov kernel and of its Fourier-Laplace perturbations. This is first done in the ergodic but non-mixing case. This work is extended by the second author to the non-ergodic case. The only prerequisites for this book are a knowledge of the basic techniques of probability theory and of notions of elementary functional analysis.
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Includes bibliographical references (pages 141-144) and index.

General facts about the method, purpose of the paper -- The central limit theorems for Markov chains -- Quasi-compact operators of diagonal type and perturbations -- First properties of Fourier kernels, application -- Peripheral eigenvalues of Fourier kernels -- Proofs of theorems A, B, C -- Renewal theorem for Markov chains (theorem D) -- Large deviations for Markov chains (theorem E) -- Ergodic properties for Markov chains -- Stochastic properties of dynamical systems -- Expanding maps -- Proofs of some statements in probability theory -- Functional analysis results on quasi-compactness -- Generalization to the non-ergodic case (by L. Herv).

This book shows how techniques from the perturbation theory of operators, applied to a quasi-compact positive kernel, may be used to obtain limit theorems for Markov chains or to describe stochastic properties of dynamical systems. A general framework for this method is given and then applied to treat several specific cases. An essential element of this work is the description of the peripheral spectra of a quasi-compact Markov kernel and of its Fourier-Laplace perturbations. This is first done in the ergodic but non-mixing case. This work is extended by the second author to the non-ergodic case. The only prerequisites for this book are a knowledge of the basic techniques of probability theory and of notions of elementary functional analysis.

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English.

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