Limit Theorems for markov chains and stochastic properties of dynamical systems by quasi-compactness

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.

• 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).