Title: Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness /
Authors: Hennion, Hubert, 1944-; Hervé, Loïc, 1963-
Publisher: Springer-Verlag,
Year: 2001
Subjects: Markov processes.; Limit theorems (Probability theory); Differentiable dynamical systems.; Stochastic processes.; Mathematics; Differential Equations.; Distribution (Probability theory).
Abstract: 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.
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. Hervé).