Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness

ISBN-13: 9783540424154 / Angielski / Miękka / 2001 / 152 str.

Hubert Hennion;Loic Herve

Opis

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.

Spis treści

General Facts About The Method Purpose Of The Paper.- The Central Limit Theorems For Markov Chains Theorems A, B, C.- Quasi-Compact Operators of Diagonal Type And Their 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.- Markov Chains Associated With Lipschitz Kernels Examples.- Stochastic Properties Of Dynamical Systems Theorems A*, B*, C*, D*, E*.- Expanding Maps.- Proofs Of Some Statements In Probability Theory.- Functional Analysis Results On Quasi-Compactness.- Generalization To The Non-Ergodic Case.